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Theorem mblfinlem1 33117
Description: Lemma for ismblfin 33121, ordering the sets of dyadic intervals that are antichains under subset and whose unions are contained entirely in 𝐴. (Contributed by Brendan Leahy, 13-Jul-2018.)
Assertion
Ref Expression
mblfinlem1 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
Distinct variable group:   𝑎,𝑏,𝑐,𝑓,𝑥,𝑦,𝐴

Proof of Theorem mblfinlem1
Dummy variables 𝑛 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 peano2re 10169 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
2 ltp1 10821 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ → 𝑛 < (𝑛 + 1))
3 breq2 4627 . . . . . . . . . . . . . 14 (𝑧 = (𝑛 + 1) → (𝑛 < 𝑧𝑛 < (𝑛 + 1)))
43rspcev 3299 . . . . . . . . . . . . 13 (((𝑛 + 1) ∈ ℝ ∧ 𝑛 < (𝑛 + 1)) → ∃𝑧 ∈ ℝ 𝑛 < 𝑧)
51, 2, 4syl2anc 692 . . . . . . . . . . . 12 (𝑛 ∈ ℝ → ∃𝑧 ∈ ℝ 𝑛 < 𝑧)
65rgen 2918 . . . . . . . . . . 11 𝑛 ∈ ℝ ∃𝑧 ∈ ℝ 𝑛 < 𝑧
7 ltnle 10077 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑛 < 𝑧 ↔ ¬ 𝑧𝑛))
87rexbidva 3044 . . . . . . . . . . . . . 14 (𝑛 ∈ ℝ → (∃𝑧 ∈ ℝ 𝑛 < 𝑧 ↔ ∃𝑧 ∈ ℝ ¬ 𝑧𝑛))
9 rexnal 2991 . . . . . . . . . . . . . 14 (∃𝑧 ∈ ℝ ¬ 𝑧𝑛 ↔ ¬ ∀𝑧 ∈ ℝ 𝑧𝑛)
108, 9syl6bb 276 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ → (∃𝑧 ∈ ℝ 𝑛 < 𝑧 ↔ ¬ ∀𝑧 ∈ ℝ 𝑧𝑛))
1110ralbiia 2975 . . . . . . . . . . . 12 (∀𝑛 ∈ ℝ ∃𝑧 ∈ ℝ 𝑛 < 𝑧 ↔ ∀𝑛 ∈ ℝ ¬ ∀𝑧 ∈ ℝ 𝑧𝑛)
12 ralnex 2988 . . . . . . . . . . . 12 (∀𝑛 ∈ ℝ ¬ ∀𝑧 ∈ ℝ 𝑧𝑛 ↔ ¬ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ℝ 𝑧𝑛)
1311, 12bitri 264 . . . . . . . . . . 11 (∀𝑛 ∈ ℝ ∃𝑧 ∈ ℝ 𝑛 < 𝑧 ↔ ¬ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ℝ 𝑧𝑛)
146, 13mpbi 220 . . . . . . . . . 10 ¬ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ℝ 𝑧𝑛
15 raleq 3131 . . . . . . . . . . 11 (𝐴 = ℝ → (∀𝑧𝐴 𝑧𝑛 ↔ ∀𝑧 ∈ ℝ 𝑧𝑛))
1615rexbidv 3047 . . . . . . . . . 10 (𝐴 = ℝ → (∃𝑛 ∈ ℝ ∀𝑧𝐴 𝑧𝑛 ↔ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ℝ 𝑧𝑛))
1714, 16mtbiri 317 . . . . . . . . 9 (𝐴 = ℝ → ¬ ∃𝑛 ∈ ℝ ∀𝑧𝐴 𝑧𝑛)
18 ssrab2 3672 . . . . . . . . . . . . . 14 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}
19 ssrab2 3672 . . . . . . . . . . . . . . 15 {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
20 zre 11341 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
21 2re 11050 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℝ
22 reexpcl 12833 . . . . . . . . . . . . . . . . . . . . 21 ((2 ∈ ℝ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℝ)
2321, 22mpan 705 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → (2↑𝑦) ∈ ℝ)
24 nn0z 11360 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℕ0𝑦 ∈ ℤ)
25 2cn 11051 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℂ
26 2ne0 11073 . . . . . . . . . . . . . . . . . . . . . 22 2 ≠ 0
27 expne0i 12848 . . . . . . . . . . . . . . . . . . . . . 22 ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ) → (2↑𝑦) ≠ 0)
2825, 26, 27mp3an12 1411 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℤ → (2↑𝑦) ≠ 0)
2924, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ ℕ0 → (2↑𝑦) ≠ 0)
3023, 29jca 554 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℕ0 → ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0))
31 redivcl 10704 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → (𝑥 / (2↑𝑦)) ∈ ℝ)
32 peano2re 10169 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
33 redivcl 10704 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 + 1) ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
3432, 33syl3an1 1356 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ)
35 opelxpi 5118 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 / (2↑𝑦)) ∈ ℝ ∧ ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
3631, 34, 35syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℝ ∧ (2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
37363expb 1263 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ ∧ ((2↑𝑦) ∈ ℝ ∧ (2↑𝑦) ≠ 0)) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
3820, 30, 37syl2an 494 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ))
3938rgen2 2971 . . . . . . . . . . . . . . . . 17 𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ)
40 eqid 2621 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
4140fmpt2 7197 . . . . . . . . . . . . . . . . 17 (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℝ × ℝ) ↔ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ))
4239, 41mpbi 220 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ)
43 frn 6020 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℝ × ℝ) → ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ (ℝ × ℝ))
4442, 43ax-mp 5 . . . . . . . . . . . . . . 15 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ (ℝ × ℝ)
4519, 44sstri 3597 . . . . . . . . . . . . . 14 {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℝ × ℝ)
4618, 45sstri 3597 . . . . . . . . . . . . 13 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ × ℝ)
47 rnss 5324 . . . . . . . . . . . . . 14 ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ × ℝ) → ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ran (ℝ × ℝ))
48 rnxpid 5536 . . . . . . . . . . . . . 14 ran (ℝ × ℝ) = ℝ
4947, 48syl6sseq 3636 . . . . . . . . . . . . 13 ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ × ℝ) → ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ℝ)
5046, 49ax-mp 5 . . . . . . . . . . . 12 ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ℝ
51 rnfi 8209 . . . . . . . . . . . 12 ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin → ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)
52 fimaxre2 10929 . . . . . . . . . . . 12 ((ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ℝ ∧ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) → ∃𝑛 ∈ ℝ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛)
5350, 51, 52sylancr 694 . . . . . . . . . . 11 ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin → ∃𝑛 ∈ ℝ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛)
5453adantl 482 . . . . . . . . . 10 (( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) → ∃𝑛 ∈ ℝ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛)
55 eluni2 4413 . . . . . . . . . . . . . . . . 17 (𝑧 ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ↔ ∃𝑢 ∈ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧𝑢)
56 iccf 12230 . . . . . . . . . . . . . . . . . . 19 [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
57 ffn 6012 . . . . . . . . . . . . . . . . . . 19 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
5856, 57ax-mp 5 . . . . . . . . . . . . . . . . . 18 [,] Fn (ℝ* × ℝ*)
59 rexpssxrxp 10044 . . . . . . . . . . . . . . . . . . 19 (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
6046, 59sstri 3597 . . . . . . . . . . . . . . . . . 18 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ* × ℝ*)
61 eleq2 2687 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ([,]‘𝑣) → (𝑧𝑢𝑧 ∈ ([,]‘𝑣)))
6261rexima 6462 . . . . . . . . . . . . . . . . . 18 (([,] Fn (ℝ* × ℝ*) ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ* × ℝ*)) → (∃𝑢 ∈ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧𝑢 ↔ ∃𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑧 ∈ ([,]‘𝑣)))
6358, 60, 62mp2an 707 . . . . . . . . . . . . . . . . 17 (∃𝑢 ∈ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧𝑢 ↔ ∃𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑧 ∈ ([,]‘𝑣))
6455, 63bitri 264 . . . . . . . . . . . . . . . 16 (𝑧 ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ↔ ∃𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑧 ∈ ([,]‘𝑣))
6546sseli 3584 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑣 ∈ (ℝ × ℝ))
66 1st2nd2 7165 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 ∈ (ℝ × ℝ) → 𝑣 = ⟨(1st𝑣), (2nd𝑣)⟩)
6766fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ (ℝ × ℝ) → ([,]‘𝑣) = ([,]‘⟨(1st𝑣), (2nd𝑣)⟩))
68 df-ov 6618 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑣)[,](2nd𝑣)) = ([,]‘⟨(1st𝑣), (2nd𝑣)⟩)
6967, 68syl6eqr 2673 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ∈ (ℝ × ℝ) → ([,]‘𝑣) = ((1st𝑣)[,](2nd𝑣)))
7069eleq2d 2684 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ (ℝ × ℝ) → (𝑧 ∈ ([,]‘𝑣) ↔ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣))))
7165, 70syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ([,]‘𝑣) ↔ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣))))
7271biimpd 219 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ([,]‘𝑣) → 𝑧 ∈ ((1st𝑣)[,](2nd𝑣))))
7372imdistani 725 . . . . . . . . . . . . . . . . . 18 ((𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ([,]‘𝑣)) → (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣))))
74 iccssxr 12214 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑣)[,](2nd𝑣)) ⊆ ℝ*
7574sseli 3584 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ((1st𝑣)[,](2nd𝑣)) → 𝑧 ∈ ℝ*)
7675ad2antll 764 . . . . . . . . . . . . . . . . . . 19 (((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣)))) → 𝑧 ∈ ℝ*)
77 xp2nd 7159 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ∈ (ℝ × ℝ) → (2nd𝑣) ∈ ℝ)
7877rexrd 10049 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ (ℝ × ℝ) → (2nd𝑣) ∈ ℝ*)
7965, 78syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (2nd𝑣) ∈ ℝ*)
8079ad2antrl 763 . . . . . . . . . . . . . . . . . . 19 (((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣)))) → (2nd𝑣) ∈ ℝ*)
81 simpllr 798 . . . . . . . . . . . . . . . . . . . 20 (((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣)))) → 𝑛 ∈ ℝ)
8281rexrd 10049 . . . . . . . . . . . . . . . . . . 19 (((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣)))) → 𝑛 ∈ ℝ*)
83 xp1st 7158 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 ∈ (ℝ × ℝ) → (1st𝑣) ∈ ℝ)
8483rexrd 10049 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ (ℝ × ℝ) → (1st𝑣) ∈ ℝ*)
8584, 78jca 554 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ∈ (ℝ × ℝ) → ((1st𝑣) ∈ ℝ* ∧ (2nd𝑣) ∈ ℝ*))
8665, 85syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ((1st𝑣) ∈ ℝ* ∧ (2nd𝑣) ∈ ℝ*))
87 iccleub 12187 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑣) ∈ ℝ* ∧ (2nd𝑣) ∈ ℝ*𝑧 ∈ ((1st𝑣)[,](2nd𝑣))) → 𝑧 ≤ (2nd𝑣))
88873expa 1262 . . . . . . . . . . . . . . . . . . . . 21 ((((1st𝑣) ∈ ℝ* ∧ (2nd𝑣) ∈ ℝ*) ∧ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣))) → 𝑧 ≤ (2nd𝑣))
8986, 88sylan 488 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣))) → 𝑧 ≤ (2nd𝑣))
9089adantl 482 . . . . . . . . . . . . . . . . . . 19 (((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣)))) → 𝑧 ≤ (2nd𝑣))
91 xpss 5197 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℝ × ℝ) ⊆ (V × V)
9246, 91sstri 3597 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (V × V)
93 df-rel 5091 . . . . . . . . . . . . . . . . . . . . . . 23 (Rel {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ↔ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (V × V))
9492, 93mpbir 221 . . . . . . . . . . . . . . . . . . . . . 22 Rel {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}
95 2ndrn 7176 . . . . . . . . . . . . . . . . . . . . . 22 ((Rel {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (2nd𝑣) ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
9694, 95mpan 705 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (2nd𝑣) ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
97 breq1 4626 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = (2nd𝑣) → (𝑢𝑛 ↔ (2nd𝑣) ≤ 𝑛))
9897rspccva 3298 . . . . . . . . . . . . . . . . . . . . 21 ((∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛 ∧ (2nd𝑣) ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (2nd𝑣) ≤ 𝑛)
9996, 98sylan2 491 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (2nd𝑣) ≤ 𝑛)
10099ad2ant2lr 783 . . . . . . . . . . . . . . . . . . 19 (((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣)))) → (2nd𝑣) ≤ 𝑛)
10176, 80, 82, 90, 100xrletrd 11953 . . . . . . . . . . . . . . . . . 18 (((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ((1st𝑣)[,](2nd𝑣)))) → 𝑧𝑛)
10273, 101sylan2 491 . . . . . . . . . . . . . . . . 17 (((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) ∧ (𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ([,]‘𝑣))) → 𝑧𝑛)
103102rexlimdvaa 3027 . . . . . . . . . . . . . . . 16 ((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) → (∃𝑣 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑧 ∈ ([,]‘𝑣) → 𝑧𝑛))
10464, 103syl5bi 232 . . . . . . . . . . . . . . 15 ((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) → (𝑧 ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → 𝑧𝑛))
105104ralrimiv 2961 . . . . . . . . . . . . . 14 ((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) → ∀𝑧 ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧𝑛)
106 raleq 3131 . . . . . . . . . . . . . . 15 ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 → (∀𝑧 ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧𝑛 ↔ ∀𝑧𝐴 𝑧𝑛))
107106ad2antrr 761 . . . . . . . . . . . . . 14 ((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) → (∀𝑧 ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})𝑧𝑛 ↔ ∀𝑧𝐴 𝑧𝑛))
108105, 107mpbid 222 . . . . . . . . . . . . 13 ((( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) ∧ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛) → ∀𝑧𝐴 𝑧𝑛)
109108ex 450 . . . . . . . . . . . 12 (( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴𝑛 ∈ ℝ) → (∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛 → ∀𝑧𝐴 𝑧𝑛))
110109reximdva 3013 . . . . . . . . . . 11 ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 → (∃𝑛 ∈ ℝ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛 → ∃𝑛 ∈ ℝ ∀𝑧𝐴 𝑧𝑛))
111110adantr 481 . . . . . . . . . 10 (( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) → (∃𝑛 ∈ ℝ ∀𝑢 ∈ ran {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}𝑢𝑛 → ∃𝑛 ∈ ℝ ∀𝑧𝐴 𝑧𝑛))
11254, 111mpd 15 . . . . . . . . 9 (( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) → ∃𝑛 ∈ ℝ ∀𝑧𝐴 𝑧𝑛)
11317, 112nsyl 135 . . . . . . . 8 (𝐴 = ℝ → ¬ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin))
114113adantl 482 . . . . . . 7 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) ∧ 𝐴 = ℝ) → ¬ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin))
115 uniretop 22506 . . . . . . . . . . 11 ℝ = (topGen‘ran (,))
116 retopconn 22572 . . . . . . . . . . . 12 (topGen‘ran (,)) ∈ Conn
117116a1i 11 . . . . . . . . . . 11 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) ∧ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → (topGen‘ran (,)) ∈ Conn)
118 simpll 789 . . . . . . . . . . 11 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) ∧ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → 𝐴 ∈ (topGen‘ran (,)))
119 simplr 791 . . . . . . . . . . 11 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) ∧ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → 𝐴 ≠ ∅)
120 simprl 793 . . . . . . . . . . . 12 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) ∧ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴)
121 ffun 6015 . . . . . . . . . . . . . . 15 ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,])
122 funiunfv 6471 . . . . . . . . . . . . . . 15 (Fun [,] → 𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
12356, 121, 122mp2b 10 . . . . . . . . . . . . . 14 𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
124 retop 22505 . . . . . . . . . . . . . . 15 (topGen‘ran (,)) ∈ Top
12546sseli 3584 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑧 ∈ (ℝ × ℝ))
126 1st2nd2 7165 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
127126fveq2d 6162 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (ℝ × ℝ) → ([,]‘𝑧) = ([,]‘⟨(1st𝑧), (2nd𝑧)⟩))
128 df-ov 6618 . . . . . . . . . . . . . . . . . . 19 ((1st𝑧)[,](2nd𝑧)) = ([,]‘⟨(1st𝑧), (2nd𝑧)⟩)
129127, 128syl6eqr 2673 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (ℝ × ℝ) → ([,]‘𝑧) = ((1st𝑧)[,](2nd𝑧)))
130 xp1st 7158 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
131 xp2nd 7159 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
132 icccld 22510 . . . . . . . . . . . . . . . . . . 19 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) → ((1st𝑧)[,](2nd𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
133130, 131, 132syl2anc 692 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧)[,](2nd𝑧)) ∈ (Clsd‘(topGen‘ran (,))))
134129, 133eqeltrd 2698 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (ℝ × ℝ) → ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran (,))))
135125, 134syl 17 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran (,))))
136135rgen 2918 . . . . . . . . . . . . . . 15 𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran (,)))
137115iuncld 20789 . . . . . . . . . . . . . . 15 (((topGen‘ran (,)) ∈ Top ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin ∧ ∀𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran (,)))) → 𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran (,))))
138124, 136, 137mp3an13 1412 . . . . . . . . . . . . . 14 ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin → 𝑧 ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran (,))))
139123, 138syl5eqelr 2703 . . . . . . . . . . . . 13 ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin → ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ∈ (Clsd‘(topGen‘ran (,))))
140139ad2antll 764 . . . . . . . . . . . 12 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) ∧ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ∈ (Clsd‘(topGen‘ran (,))))
141120, 140eqeltrrd 2699 . . . . . . . . . . 11 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) ∧ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → 𝐴 ∈ (Clsd‘(topGen‘ran (,))))
142115, 117, 118, 119, 141connclo 21158 . . . . . . . . . 10 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) ∧ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)) → 𝐴 = ℝ)
143142ex 450 . . . . . . . . 9 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → (( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin) → 𝐴 = ℝ))
144143necon3ad 2803 . . . . . . . 8 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → (𝐴 ≠ ℝ → ¬ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)))
145144imp 445 . . . . . . 7 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) ∧ 𝐴 ≠ ℝ) → ¬ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin))
146114, 145pm2.61dane 2877 . . . . . 6 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → ¬ ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin))
147 oveq1 6622 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (𝑥 / (2↑𝑦)) = (𝑢 / (2↑𝑦)))
148 oveq1 6622 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → (𝑥 + 1) = (𝑢 + 1))
149148oveq1d 6630 . . . . . . . . . . . 12 (𝑥 = 𝑢 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑦)))
150147, 149opeq12d 4385 . . . . . . . . . . 11 (𝑥 = 𝑢 → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩)
151 oveq2 6623 . . . . . . . . . . . . 13 (𝑦 = 𝑣 → (2↑𝑦) = (2↑𝑣))
152151oveq2d 6631 . . . . . . . . . . . 12 (𝑦 = 𝑣 → (𝑢 / (2↑𝑦)) = (𝑢 / (2↑𝑣)))
153151oveq2d 6631 . . . . . . . . . . . 12 (𝑦 = 𝑣 → ((𝑢 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑣)))
154152, 153opeq12d 4385 . . . . . . . . . . 11 (𝑦 = 𝑣 → ⟨(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))⟩ = ⟨(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))⟩)
155150, 154cbvmpt2v 6700 . . . . . . . . . 10 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) = (𝑢 ∈ ℤ, 𝑣 ∈ ℕ0 ↦ ⟨(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))⟩)
156 fveq2 6158 . . . . . . . . . . . . . 14 (𝑎 = 𝑧 → ([,]‘𝑎) = ([,]‘𝑧))
157156sseq1d 3617 . . . . . . . . . . . . 13 (𝑎 = 𝑧 → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘𝑧) ⊆ ([,]‘𝑐)))
158 equequ1 1949 . . . . . . . . . . . . 13 (𝑎 = 𝑧 → (𝑎 = 𝑐𝑧 = 𝑐))
159157, 158imbi12d 334 . . . . . . . . . . . 12 (𝑎 = 𝑧 → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)))
160159ralbidv 2982 . . . . . . . . . . 11 (𝑎 = 𝑧 → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)))
161160cbvrabv 3189 . . . . . . . . . 10 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} = {𝑧 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)}
16219a1i 11 . . . . . . . . . 10 (𝐴 ∈ (topGen‘ran (,)) → {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
163155, 161, 162dyadmbllem 23307 . . . . . . . . 9 (𝐴 ∈ (topGen‘ran (,)) → ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}))
164 opnmbllem0 33116 . . . . . . . . 9 (𝐴 ∈ (topGen‘ran (,)) → ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴)
165163, 164eqtr3d 2657 . . . . . . . 8 (𝐴 ∈ (topGen‘ran (,)) → ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴)
166165adantr 481 . . . . . . 7 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴)
167 nnenom 12735 . . . . . . . . 9 ℕ ≈ ω
168 sdomentr 8054 . . . . . . . . 9 (({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ ∧ ℕ ≈ ω) → {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ω)
169167, 168mpan2 706 . . . . . . . 8 ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ → {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ω)
170 isfinite 8509 . . . . . . . 8 ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin ↔ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ω)
171169, 170sylibr 224 . . . . . . 7 ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ → {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin)
172166, 171anim12i 589 . . . . . 6 (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ) → ( ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) = 𝐴 ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∈ Fin))
173146, 172mtand 690 . . . . 5 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → ¬ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ)
174 qex 11760 . . . . . . . 8 ℚ ∈ V
175174, 174xpex 6927 . . . . . . 7 (ℚ × ℚ) ∈ V
176 zq 11754 . . . . . . . . . . . . 13 (𝑥 ∈ ℤ → 𝑥 ∈ ℚ)
177 2nn 11145 . . . . . . . . . . . . . . . 16 2 ∈ ℕ
178 nnq 11761 . . . . . . . . . . . . . . . 16 (2 ∈ ℕ → 2 ∈ ℚ)
179177, 178ax-mp 5 . . . . . . . . . . . . . . 15 2 ∈ ℚ
180 qexpcl 12832 . . . . . . . . . . . . . . 15 ((2 ∈ ℚ ∧ 𝑦 ∈ ℕ0) → (2↑𝑦) ∈ ℚ)
181179, 180mpan 705 . . . . . . . . . . . . . 14 (𝑦 ∈ ℕ0 → (2↑𝑦) ∈ ℚ)
182181, 29jca 554 . . . . . . . . . . . . 13 (𝑦 ∈ ℕ0 → ((2↑𝑦) ∈ ℚ ∧ (2↑𝑦) ≠ 0))
183 qdivcl 11769 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℚ ∧ (2↑𝑦) ∈ ℚ ∧ (2↑𝑦) ≠ 0) → (𝑥 / (2↑𝑦)) ∈ ℚ)
184 1z 11367 . . . . . . . . . . . . . . . . . 18 1 ∈ ℤ
185 zq 11754 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℤ → 1 ∈ ℚ)
186184, 185ax-mp 5 . . . . . . . . . . . . . . . . 17 1 ∈ ℚ
187 qaddcl 11764 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℚ ∧ 1 ∈ ℚ) → (𝑥 + 1) ∈ ℚ)
188186, 187mpan2 706 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℚ → (𝑥 + 1) ∈ ℚ)
189 qdivcl 11769 . . . . . . . . . . . . . . . 16 (((𝑥 + 1) ∈ ℚ ∧ (2↑𝑦) ∈ ℚ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℚ)
190188, 189syl3an1 1356 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℚ ∧ (2↑𝑦) ∈ ℚ ∧ (2↑𝑦) ≠ 0) → ((𝑥 + 1) / (2↑𝑦)) ∈ ℚ)
191 opelxpi 5118 . . . . . . . . . . . . . . 15 (((𝑥 / (2↑𝑦)) ∈ ℚ ∧ ((𝑥 + 1) / (2↑𝑦)) ∈ ℚ) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℚ × ℚ))
192183, 190, 191syl2anc 692 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℚ ∧ (2↑𝑦) ∈ ℚ ∧ (2↑𝑦) ≠ 0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℚ × ℚ))
1931923expb 1263 . . . . . . . . . . . . 13 ((𝑥 ∈ ℚ ∧ ((2↑𝑦) ∈ ℚ ∧ (2↑𝑦) ≠ 0)) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℚ × ℚ))
194176, 182, 193syl2an 494 . . . . . . . . . . . 12 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0) → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℚ × ℚ))
195194rgen2 2971 . . . . . . . . . . 11 𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℚ × ℚ)
19640fmpt2 7197 . . . . . . . . . . 11 (∀𝑥 ∈ ℤ ∀𝑦 ∈ ℕ0 ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ ∈ (ℚ × ℚ) ↔ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℚ × ℚ))
197195, 196mpbi 220 . . . . . . . . . 10 (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℚ × ℚ)
198 frn 6020 . . . . . . . . . 10 ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ0)⟶(ℚ × ℚ) → ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ (ℚ × ℚ))
199197, 198ax-mp 5 . . . . . . . . 9 ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ (ℚ × ℚ)
20019, 199sstri 3597 . . . . . . . 8 {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℚ × ℚ)
20118, 200sstri 3597 . . . . . . 7 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℚ × ℚ)
202 ssdomg 7961 . . . . . . 7 ((ℚ × ℚ) ∈ V → ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℚ × ℚ) → {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ (ℚ × ℚ)))
203175, 201, 202mp2 9 . . . . . 6 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ (ℚ × ℚ)
204 qnnen 14886 . . . . . . . 8 ℚ ≈ ℕ
205 xpen 8083 . . . . . . . 8 ((ℚ ≈ ℕ ∧ ℚ ≈ ℕ) → (ℚ × ℚ) ≈ (ℕ × ℕ))
206204, 204, 205mp2an 707 . . . . . . 7 (ℚ × ℚ) ≈ (ℕ × ℕ)
207 xpnnen 14883 . . . . . . 7 (ℕ × ℕ) ≈ ℕ
208206, 207entri 7970 . . . . . 6 (ℚ × ℚ) ≈ ℕ
209 domentr 7975 . . . . . 6 (({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ (ℚ × ℚ) ∧ (ℚ × ℚ) ≈ ℕ) → {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ ℕ)
210203, 208, 209mp2an 707 . . . . 5 {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ ℕ
211173, 210jctil 559 . . . 4 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ ℕ ∧ ¬ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ))
212 bren2 7946 . . . 4 ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≈ ℕ ↔ ({𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≼ ℕ ∧ ¬ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≺ ℕ))
213211, 212sylibr 224 . . 3 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ≈ ℕ)
214213ensymd 7967 . 2 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → ℕ ≈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
215 bren 7924 . 2 (ℕ ≈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ↔ ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
216214, 215sylib 208 1 ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2908  wrex 2909  {crab 2912  Vcvv 3190  wss 3560  c0 3897  𝒫 cpw 4136  cop 4161   cuni 4409   ciun 4492   class class class wbr 4623   × cxp 5082  ran crn 5085  cima 5087  Rel wrel 5089  Fun wfun 5851   Fn wfn 5852  wf 5853  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  cmpt2 6617  ωcom 7027  1st c1st 7126  2nd c2nd 7127  cen 7912  cdom 7913  csdm 7914  Fincfn 7915  cc 9894  cr 9895  0cc0 9896  1c1 9897   + caddc 9899  *cxr 10033   < clt 10034  cle 10035   / cdiv 10644  cn 10980  2c2 11030  0cn0 11252  cz 11337  cq 11748  (,)cioo 12133  [,]cicc 12136  cexp 12816  topGenctg 16038  Topctop 20638  Clsdccld 20760  Conncconn 21154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fi 8277  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-acn 8728  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-sum 14367  df-rest 16023  df-topgen 16044  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-top 20639  df-topon 20656  df-bases 20690  df-cld 20763  df-cmp 21130  df-conn 21155  df-ovol 23173
This theorem is referenced by:  mblfinlem2  33118
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