Step | Hyp | Ref
| Expression |
1 | | peano2re 10421 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℝ → (𝑛 + 1) ∈
ℝ) |
2 | | ltp1 11073 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℝ → 𝑛 < (𝑛 + 1)) |
3 | | breq2 4808 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑛 + 1) → (𝑛 < 𝑧 ↔ 𝑛 < (𝑛 + 1))) |
4 | 3 | rspcev 3449 |
. . . . . . . . . . . . 13
⊢ (((𝑛 + 1) ∈ ℝ ∧ 𝑛 < (𝑛 + 1)) → ∃𝑧 ∈ ℝ 𝑛 < 𝑧) |
5 | 1, 2, 4 | syl2anc 696 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℝ →
∃𝑧 ∈ ℝ
𝑛 < 𝑧) |
6 | 5 | rgen 3060 |
. . . . . . . . . . 11
⊢
∀𝑛 ∈
ℝ ∃𝑧 ∈
ℝ 𝑛 < 𝑧 |
7 | | ltnle 10329 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑛 < 𝑧 ↔ ¬ 𝑧 ≤ 𝑛)) |
8 | 7 | rexbidva 3187 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℝ →
(∃𝑧 ∈ ℝ
𝑛 < 𝑧 ↔ ∃𝑧 ∈ ℝ ¬ 𝑧 ≤ 𝑛)) |
9 | | rexnal 3133 |
. . . . . . . . . . . . . 14
⊢
(∃𝑧 ∈
ℝ ¬ 𝑧 ≤ 𝑛 ↔ ¬ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛) |
10 | 8, 9 | syl6bb 276 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℝ →
(∃𝑧 ∈ ℝ
𝑛 < 𝑧 ↔ ¬ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛)) |
11 | 10 | ralbiia 3117 |
. . . . . . . . . . . 12
⊢
(∀𝑛 ∈
ℝ ∃𝑧 ∈
ℝ 𝑛 < 𝑧 ↔ ∀𝑛 ∈ ℝ ¬
∀𝑧 ∈ ℝ
𝑧 ≤ 𝑛) |
12 | | ralnex 3130 |
. . . . . . . . . . . 12
⊢
(∀𝑛 ∈
ℝ ¬ ∀𝑧
∈ ℝ 𝑧 ≤ 𝑛 ↔ ¬ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛) |
13 | 11, 12 | bitri 264 |
. . . . . . . . . . 11
⊢
(∀𝑛 ∈
ℝ ∃𝑧 ∈
ℝ 𝑛 < 𝑧 ↔ ¬ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛) |
14 | 6, 13 | mpbi 220 |
. . . . . . . . . 10
⊢ ¬
∃𝑛 ∈ ℝ
∀𝑧 ∈ ℝ
𝑧 ≤ 𝑛 |
15 | | raleq 3277 |
. . . . . . . . . . 11
⊢ (𝐴 = ℝ → (∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛 ↔ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛)) |
16 | 15 | rexbidv 3190 |
. . . . . . . . . 10
⊢ (𝐴 = ℝ → (∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛 ↔ ∃𝑛 ∈ ℝ ∀𝑧 ∈ ℝ 𝑧 ≤ 𝑛)) |
17 | 14, 16 | mtbiri 316 |
. . . . . . . . 9
⊢ (𝐴 = ℝ → ¬
∃𝑛 ∈ ℝ
∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛) |
18 | | ssrab2 3828 |
. . . . . . . . . . . . . 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 3828 |
. . . . . . . . . . . . . . 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 11593 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ℤ → 𝑥 ∈
ℝ) |
21 | | 2re 11302 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ∈
ℝ |
22 | | reexpcl 13091 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((2
∈ ℝ ∧ 𝑦
∈ ℕ0) → (2↑𝑦) ∈ ℝ) |
23 | 21, 22 | mpan 708 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ℕ0
→ (2↑𝑦) ∈
ℝ) |
24 | | nn0z 11612 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℕ0
→ 𝑦 ∈
ℤ) |
25 | | 2cn 11303 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
ℂ |
26 | | 2ne0 11325 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ≠
0 |
27 | | expne0i 13106 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((2
∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ) → (2↑𝑦) ≠ 0) |
28 | 25, 26, 27 | mp3an12 1563 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℤ →
(2↑𝑦) ≠
0) |
29 | 24, 28 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ℕ0
→ (2↑𝑦) ≠
0) |
30 | 23, 29 | jca 555 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ ℕ0
→ ((2↑𝑦) ∈
ℝ ∧ (2↑𝑦)
≠ 0)) |
31 | | redivcl 10956 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ (𝑥 / (2↑𝑦)) ∈
ℝ) |
32 | | peano2re 10421 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ) |
33 | | redivcl 10956 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑥 + 1) ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℝ) |
34 | 32, 33 | syl3an1 1167 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℝ) |
35 | | opelxpi 5305 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑥 / (2↑𝑦)) ∈ ℝ ∧ ((𝑥 + 1) / (2↑𝑦)) ∈ ℝ) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
36 | 31, 34, 35 | syl2anc 696 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
37 | 36 | 3expb 1114 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ ℝ ∧
((2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0))
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
38 | 20, 30, 37 | syl2an 495 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
39 | 38 | rgen2 3113 |
. . . . . . . . . . . . . . . . 17
⊢
∀𝑥 ∈
ℤ ∀𝑦 ∈
ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ) |
40 | | eqid 2760 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
41 | 40 | fmpt2 7406 |
. . . . . . . . . . . . . . . . 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)) |
42 | 39, 41 | mpbi 220 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℝ × ℝ) |
43 | | frn 6214 |
. . . . . . . . . . . . . . . 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)) |
44 | 42, 43 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ (ℝ
× ℝ) |
45 | 19, 44 | sstri 3753 |
. . . . . . . . . . . . . 14
⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℝ ×
ℝ) |
46 | 18, 45 | sstri 3753 |
. . . . . . . . . . . . 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 5509 |
. . . . . . . . . . . . . 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 5725 |
. . . . . . . . . . . . . 14
⊢ ran
(ℝ × ℝ) = ℝ |
49 | 47, 48 | syl6sseq 3792 |
. . . . . . . . . . . . 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) |
50 | 46, 49 | ax-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 8416 |
. . . . . . . . . . . 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 11181 |
. . . . . . . . . . . 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)}𝑢 ≤ 𝑛) |
53 | 50, 51, 52 | sylancr 698 |
. . . . . . . . . . 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)}𝑢 ≤ 𝑛) |
54 | 53 | adantl 473 |
. . . . . . . . . 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)}𝑢 ≤ 𝑛) |
55 | | eluni2 4592 |
. . . . . . . . . . . . . . . . 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)})𝑧 ∈ 𝑢) |
56 | | iccf 12485 |
. . . . . . . . . . . . . . . . . . 19
⊢
[,]:(ℝ* × ℝ*)⟶𝒫
ℝ* |
57 | | ffn 6206 |
. . . . . . . . . . . . . . . . . . 19
⊢
([,]:(ℝ* × ℝ*)⟶𝒫
ℝ* → [,] Fn (ℝ* ×
ℝ*)) |
58 | 56, 57 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ [,] Fn
(ℝ* × ℝ*) |
59 | | rexpssxrxp 10296 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℝ
× ℝ) ⊆ (ℝ* ×
ℝ*) |
60 | 46, 59 | sstri 3753 |
. . . . . . . . . . . . . . . . . 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 2828 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑢 = ([,]‘𝑣) → (𝑧 ∈ 𝑢 ↔ 𝑧 ∈ ([,]‘𝑣))) |
62 | 61 | rexima 6661 |
. . . . . . . . . . . . . . . . . 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)})𝑧 ∈ 𝑢 ↔ ∃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))) |
63 | 58, 60, 62 | mp2an 710 |
. . . . . . . . . . . . . . . . 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)})𝑧 ∈ 𝑢 ↔ ∃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)) |
64 | 55, 63 | bitri 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)) |
65 | 46 | sseli 3740 |
. . . . . . . . . . . . . . . . . . . . 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 7373 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑣 ∈ (ℝ ×
ℝ) → 𝑣 =
〈(1st ‘𝑣), (2nd ‘𝑣)〉) |
67 | 66 | fveq2d 6357 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑣 ∈ (ℝ ×
ℝ) → ([,]‘𝑣) = ([,]‘〈(1st
‘𝑣), (2nd
‘𝑣)〉)) |
68 | | df-ov 6817 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((1st ‘𝑣)[,](2nd ‘𝑣)) = ([,]‘〈(1st
‘𝑣), (2nd
‘𝑣)〉) |
69 | 67, 68 | syl6eqr 2812 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 ∈ (ℝ ×
ℝ) → ([,]‘𝑣) = ((1st ‘𝑣)[,](2nd ‘𝑣))) |
70 | 69 | eleq2d 2825 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 ∈ (ℝ ×
ℝ) → (𝑧 ∈
([,]‘𝑣) ↔ 𝑧 ∈ ((1st
‘𝑣)[,](2nd
‘𝑣)))) |
71 | 65, 70 | syl 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 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) |
72 | 71 | biimpd 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 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) |
73 | 72 | imdistani 728 |
. . . . . . . . . . . . . . . . . 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 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) |
74 | | iccssxr 12469 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘𝑣)[,](2nd ‘𝑣)) ⊆
ℝ* |
75 | 74 | sseli 3740 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ((1st
‘𝑣)[,](2nd
‘𝑣)) → 𝑧 ∈
ℝ*) |
76 | 75 | ad2antll 767 |
. . . . . . . . . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛) ∧ (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 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → 𝑧 ∈ ℝ*) |
77 | | xp2nd 7367 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 ∈ (ℝ ×
ℝ) → (2nd ‘𝑣) ∈ ℝ) |
78 | 77 | rexrd 10301 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 ∈ (ℝ ×
ℝ) → (2nd ‘𝑣) ∈
ℝ*) |
79 | 65, 78 | syl 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)} → (2nd ‘𝑣) ∈
ℝ*) |
80 | 79 | ad2antrl 766 |
. . . . . . . . . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛) ∧ (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 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → (2nd
‘𝑣) ∈
ℝ*) |
81 | | simpllr 817 |
. . . . . . . . . . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛) ∧ (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 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → 𝑛 ∈ ℝ) |
82 | 81 | rexrd 10301 |
. . . . . . . . . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛) ∧ (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 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → 𝑛 ∈ ℝ*) |
83 | | xp1st 7366 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑣 ∈ (ℝ ×
ℝ) → (1st ‘𝑣) ∈ ℝ) |
84 | 83 | rexrd 10301 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑣 ∈ (ℝ ×
ℝ) → (1st ‘𝑣) ∈
ℝ*) |
85 | 84, 78 | jca 555 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 ∈ (ℝ ×
ℝ) → ((1st ‘𝑣) ∈ ℝ* ∧
(2nd ‘𝑣)
∈ ℝ*)) |
86 | 65, 85 | syl 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)} → ((1st ‘𝑣) ∈ ℝ*
∧ (2nd ‘𝑣) ∈
ℝ*)) |
87 | | iccleub 12442 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((1st ‘𝑣) ∈ ℝ* ∧
(2nd ‘𝑣)
∈ ℝ* ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣))) → 𝑧 ≤ (2nd ‘𝑣)) |
88 | 87 | 3expa 1112 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((1st ‘𝑣) ∈ ℝ* ∧
(2nd ‘𝑣)
∈ ℝ*) ∧ 𝑧 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣))) → 𝑧 ≤ (2nd ‘𝑣)) |
89 | 86, 88 | sylan 489 |
. . . . . . . . . . . . . . . . . . . 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 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣))) → 𝑧 ≤ (2nd ‘𝑣)) |
90 | 89 | adantl 473 |
. . . . . . . . . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛) ∧ (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 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → 𝑧 ≤ (2nd ‘𝑣)) |
91 | | xpss 5282 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (ℝ
× ℝ) ⊆ (V × V) |
92 | 46, 91 | sstri 3753 |
. . . . . . . . . . . . . . . . . . . . . . 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 5273 |
. . . . . . . . . . . . . . . . . . . . . . 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)) |
94 | 92, 93 | mpbir 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 7384 |
. . . . . . . . . . . . . . . . . . . . . 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)}) → (2nd ‘𝑣) ∈ 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)}) |
96 | 94, 95 | mpan 708 |
. . . . . . . . . . . . . . . . . . . . 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)} → (2nd ‘𝑣) ∈ 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 4807 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑢 = (2nd ‘𝑣) → (𝑢 ≤ 𝑛 ↔ (2nd ‘𝑣) ≤ 𝑛)) |
98 | 97 | rspccva 3448 |
. . . . . . . . . . . . . . . . . . . . 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)}𝑢 ≤ 𝑛 ∧ (2nd ‘𝑣) ∈ 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)}) → (2nd ‘𝑣) ≤ 𝑛) |
99 | 96, 98 | sylan2 492 |
. . . . . . . . . . . . . . . . . . . 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)}𝑢 ≤ 𝑛 ∧ 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)}) → (2nd ‘𝑣) ≤ 𝑛) |
100 | 99 | ad2ant2lr 801 |
. . . . . . . . . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛) ∧ (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 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → (2nd
‘𝑣) ≤ 𝑛) |
101 | 76, 80, 82, 90, 100 | xrletrd 12206 |
. . . . . . . . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛) ∧ (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 ∈ ((1st ‘𝑣)[,](2nd ‘𝑣)))) → 𝑧 ≤ 𝑛) |
102 | 73, 101 | sylan2 492 |
. . . . . . . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛) ∧ (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 ≤ 𝑛) |
103 | 102 | rexlimdvaa 3170 |
. . . . . . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛) → (∃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 ≤ 𝑛)) |
104 | 64, 103 | syl5bi 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)}) = 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)}𝑢 ≤ 𝑛) → (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 ≤ 𝑛)) |
105 | 104 | ralrimiv 3103 |
. . . . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛) → ∀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 ≤ 𝑛) |
106 | | raleq 3277 |
. . . . . . . . . . . . . . 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)})𝑧 ≤ 𝑛 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛)) |
107 | 106 | ad2antrr 764 |
. . . . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛) → (∀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 ≤ 𝑛 ↔ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛)) |
108 | 105, 107 | mpbid 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)}) = 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)}𝑢 ≤ 𝑛) → ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛) |
109 | 108 | ex 449 |
. . . . . . . . . . . 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)}) = 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)}𝑢 ≤ 𝑛 → ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛)) |
110 | 109 | reximdva 3155 |
. . . . . . . . . . 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)}𝑢 ≤ 𝑛 → ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛)) |
111 | 110 | adantr 472 |
. . . . . . . . . 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)}𝑢 ≤ 𝑛 → ∃𝑛 ∈ ℝ ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛)) |
112 | 54, 111 | mpd 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 ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝑛) |
113 | 17, 112 | nsyl 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)) |
114 | 113 | adantl 473 |
. . . . . . 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 22787 |
. . . . . . . . . . 11
⊢ ℝ =
∪ (topGen‘ran (,)) |
116 | | retopconn 22853 |
. . . . . . . . . . . 12
⊢
(topGen‘ran (,)) ∈ Conn |
117 | 116 | a1i 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 807 |
. . . . . . . . . . 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 809 |
. . . . . . . . . . 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 811 |
. . . . . . . . . . . 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 6209 |
. . . . . . . . . . . . . . 15
⊢
([,]:(ℝ* × ℝ*)⟶𝒫
ℝ* → Fun [,]) |
122 | | funiunfv 6670 |
. . . . . . . . . . . . . . 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)})) |
123 | 56, 121, 122 | mp2b 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 22786 |
. . . . . . . . . . . . . . 15
⊢
(topGen‘ran (,)) ∈ Top |
125 | 46 | sseli 3740 |
. . . . . . . . . . . . . . . . 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 7373 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ (ℝ ×
ℝ) → 𝑧 =
〈(1st ‘𝑧), (2nd ‘𝑧)〉) |
127 | 126 | fveq2d 6357 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ (ℝ ×
ℝ) → ([,]‘𝑧) = ([,]‘〈(1st
‘𝑧), (2nd
‘𝑧)〉)) |
128 | | df-ov 6817 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑧)[,](2nd ‘𝑧)) = ([,]‘〈(1st
‘𝑧), (2nd
‘𝑧)〉) |
129 | 127, 128 | syl6eqr 2812 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ (ℝ ×
ℝ) → ([,]‘𝑧) = ((1st ‘𝑧)[,](2nd ‘𝑧))) |
130 | | xp1st 7366 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ (ℝ ×
ℝ) → (1st ‘𝑧) ∈ ℝ) |
131 | | xp2nd 7367 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ (ℝ ×
ℝ) → (2nd ‘𝑧) ∈ ℝ) |
132 | | icccld 22791 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1st ‘𝑧) ∈ ℝ ∧ (2nd
‘𝑧) ∈ ℝ)
→ ((1st ‘𝑧)[,](2nd ‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
133 | 130, 131,
132 | syl2anc 696 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ (ℝ ×
ℝ) → ((1st ‘𝑧)[,](2nd ‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
134 | 129, 133 | eqeltrd 2839 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 ∈ (ℝ ×
ℝ) → ([,]‘𝑧) ∈ (Clsd‘(topGen‘ran
(,)))) |
135 | 125, 134 | syl 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
(,)))) |
136 | 135 | rgen 3060 |
. . . . . . . . . . . . . . 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
(,))) |
137 | 115 | iuncld 21071 |
. . . . . . . . . . . . . . 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
(,)))) |
138 | 124, 136,
137 | mp3an13 1564 |
. . . . . . . . . . . . . 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
(,)))) |
139 | 123, 138 | syl5eqelr 2844 |
. . . . . . . . . . . . 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
(,)))) |
140 | 139 | ad2antll 767 |
. . . . . . . . . . . 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
(,)))) |
141 | 120, 140 | eqeltrrd 2840 |
. . . . . . . . . . 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
(,)))) |
142 | 115, 117,
118, 119, 141 | connclo 21440 |
. . . . . . . . . 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) |
143 | 142 | ex 449 |
. . . . . . . . 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)) |
144 | 143 | necon3ad 2945 |
. . . . . . . 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))) |
145 | 144 | imp 444 |
. . . . . . 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)) |
146 | 114, 145 | pm2.61dane 3019 |
. . . . . 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 6821 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑢 → (𝑥 / (2↑𝑦)) = (𝑢 / (2↑𝑦))) |
148 | | oveq1 6821 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑢 → (𝑥 + 1) = (𝑢 + 1)) |
149 | 148 | oveq1d 6829 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑢 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑦))) |
150 | 147, 149 | opeq12d 4561 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑢 → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 = 〈(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))〉) |
151 | | oveq2 6822 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑣 → (2↑𝑦) = (2↑𝑣)) |
152 | 151 | oveq2d 6830 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑣 → (𝑢 / (2↑𝑦)) = (𝑢 / (2↑𝑣))) |
153 | 151 | oveq2d 6830 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑣 → ((𝑢 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑣))) |
154 | 152, 153 | opeq12d 4561 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑣 → 〈(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))〉 = 〈(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))〉) |
155 | 150, 154 | cbvmpt2v 6901 |
. . . . . . . . . 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 6353 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑧 → ([,]‘𝑎) = ([,]‘𝑧)) |
157 | 156 | sseq1d 3773 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑧 → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘𝑧) ⊆ ([,]‘𝑐))) |
158 | | equequ1 2107 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑧 → (𝑎 = 𝑐 ↔ 𝑧 = 𝑐)) |
159 | 157, 158 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑧 → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐))) |
160 | 159 | ralbidv 3124 |
. . . . . . . . . . 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))) |
161 | 160 | cbvrabv 3339 |
. . . . . . . . . 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)} |
162 | 19 | a1i 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))〉)) |
163 | 155, 161,
162 | dyadmbllem 23587 |
. . . . . . . . 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 33776 |
. . . . . . . . 9
⊢ (𝐴 ∈ (topGen‘ran (,))
→ ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴) |
165 | 163, 164 | eqtr3d 2796 |
. . . . . . . 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) |
166 | 165 | adantr 472 |
. . . . . . 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 12993 |
. . . . . . . . 9
⊢ ℕ
≈ ω |
168 | | sdomentr 8261 |
. . . . . . . . 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) |
169 | 167, 168 | mpan2 709 |
. . . . . . . 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 8724 |
. . . . . . . 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) |
171 | 169, 170 | sylibr 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) |
172 | 166, 171 | anim12i 591 |
. . . . . 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)) |
173 | 146, 172 | mtand 694 |
. . . . 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 12013 |
. . . . . . . 8
⊢ ℚ
∈ V |
175 | 174, 174 | xpex 7128 |
. . . . . . 7
⊢ (ℚ
× ℚ) ∈ V |
176 | | zq 12007 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℤ → 𝑥 ∈
ℚ) |
177 | | 2nn 11397 |
. . . . . . . . . . . . . . . 16
⊢ 2 ∈
ℕ |
178 | | nnq 12014 |
. . . . . . . . . . . . . . . 16
⊢ (2 ∈
ℕ → 2 ∈ ℚ) |
179 | 177, 178 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℚ |
180 | | qexpcl 13090 |
. . . . . . . . . . . . . . 15
⊢ ((2
∈ ℚ ∧ 𝑦
∈ ℕ0) → (2↑𝑦) ∈ ℚ) |
181 | 179, 180 | mpan 708 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℕ0
→ (2↑𝑦) ∈
ℚ) |
182 | 181, 29 | jca 555 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ0
→ ((2↑𝑦) ∈
ℚ ∧ (2↑𝑦)
≠ 0)) |
183 | | qdivcl 12022 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℚ ∧
(2↑𝑦) ∈ ℚ
∧ (2↑𝑦) ≠ 0)
→ (𝑥 / (2↑𝑦)) ∈
ℚ) |
184 | | 1z 11619 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℤ |
185 | | zq 12007 |
. . . . . . . . . . . . . . . . . 18
⊢ (1 ∈
ℤ → 1 ∈ ℚ) |
186 | 184, 185 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℚ |
187 | | qaddcl 12017 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℚ ∧ 1 ∈
ℚ) → (𝑥 + 1)
∈ ℚ) |
188 | 186, 187 | mpan2 709 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℚ → (𝑥 + 1) ∈
ℚ) |
189 | | qdivcl 12022 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 + 1) ∈ ℚ ∧
(2↑𝑦) ∈ ℚ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℚ) |
190 | 188, 189 | syl3an1 1167 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℚ ∧
(2↑𝑦) ∈ ℚ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℚ) |
191 | | opelxpi 5305 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 / (2↑𝑦)) ∈ ℚ ∧ ((𝑥 + 1) / (2↑𝑦)) ∈ ℚ) → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℚ ×
ℚ)) |
192 | 183, 190,
191 | syl2anc 696 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℚ ∧
(2↑𝑦) ∈ ℚ
∧ (2↑𝑦) ≠ 0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℚ ×
ℚ)) |
193 | 192 | 3expb 1114 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℚ ∧
((2↑𝑦) ∈ ℚ
∧ (2↑𝑦) ≠ 0))
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℚ ×
ℚ)) |
194 | 176, 182,
193 | syl2an 495 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℚ ×
ℚ)) |
195 | 194 | rgen2 3113 |
. . . . . . . . . . 11
⊢
∀𝑥 ∈
ℤ ∀𝑦 ∈
ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℚ ×
ℚ) |
196 | 40 | fmpt2 7406 |
. . . . . . . . . . 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)) |
197 | 195, 196 | mpbi 220 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℚ × ℚ) |
198 | | frn 6214 |
. . . . . . . . . 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)) |
199 | 197, 198 | ax-mp 5 |
. . . . . . . . 9
⊢ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ (ℚ
× ℚ) |
200 | 19, 199 | sstri 3753 |
. . . . . . . 8
⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℚ ×
ℚ) |
201 | 18, 200 | sstri 3753 |
. . . . . . 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 8169 |
. . . . . . 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))) |
203 | 175, 201,
202 | mp2 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 15161 |
. . . . . . . 8
⊢ ℚ
≈ ℕ |
205 | | xpen 8290 |
. . . . . . . 8
⊢ ((ℚ
≈ ℕ ∧ ℚ ≈ ℕ) → (ℚ × ℚ)
≈ (ℕ × ℕ)) |
206 | 204, 204,
205 | mp2an 710 |
. . . . . . 7
⊢ (ℚ
× ℚ) ≈ (ℕ × ℕ) |
207 | | xpnnen 15158 |
. . . . . . 7
⊢ (ℕ
× ℕ) ≈ ℕ |
208 | 206, 207 | entri 8177 |
. . . . . 6
⊢ (ℚ
× ℚ) ≈ ℕ |
209 | | domentr 8182 |
. . . . . 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) |
210 | 203, 208,
209 | mp2an 710 |
. . . . 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 |
211 | 173, 210 | jctil 561 |
. . . 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 8154 |
. . . 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)) |
213 | 211, 212 | sylibr 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) |
214 | 213 | ensymd 8174 |
. 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 8132 |
. 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)}) |
216 | 214, 215 | sylib 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)}) |