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Theorem cctop 21030
Description: The countable complement topology on a set 𝐴. Example 4 in [Munkres] p. 77. (Contributed by FL, 23-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
cctop (𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem cctop
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniss 4593 . . . . . . . 8 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
2 ssrab2 3834 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴
3 sspwuni 4743 . . . . . . . . 9 ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴)
42, 3mpbi 220 . . . . . . . 8 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴
51, 4syl6ss 3762 . . . . . . 7 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦𝐴)
6 vuniex 7100 . . . . . . . 8 𝑦 ∈ V
76elpw 4301 . . . . . . 7 ( 𝑦 ∈ 𝒫 𝐴 𝑦𝐴)
85, 7sylibr 224 . . . . . 6 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ 𝒫 𝐴)
9 uni0c 4598 . . . . . . . . . . 11 ( 𝑦 = ∅ ↔ ∀𝑧𝑦 𝑧 = ∅)
109notbii 309 . . . . . . . . . 10 𝑦 = ∅ ↔ ¬ ∀𝑧𝑦 𝑧 = ∅)
11 rexnal 3142 . . . . . . . . . 10 (∃𝑧𝑦 ¬ 𝑧 = ∅ ↔ ¬ ∀𝑧𝑦 𝑧 = ∅)
1210, 11bitr4i 267 . . . . . . . . 9 𝑦 = ∅ ↔ ∃𝑧𝑦 ¬ 𝑧 = ∅)
13 ssel2 3745 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
14 difeq2 3871 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑧 → (𝐴𝑥) = (𝐴𝑧))
1514breq1d 4794 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → ((𝐴𝑥) ≼ ω ↔ (𝐴𝑧) ≼ ω))
16 eqeq1 2774 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
1715, 16orbi12d 883 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
1817elrab 3513 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
1913, 18sylib 208 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
2019simprd 477 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))
2120ord 844 . . . . . . . . . . . . . 14 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (¬ (𝐴𝑧) ≼ ω → 𝑧 = ∅))
2221con1d 141 . . . . . . . . . . . . 13 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (¬ 𝑧 = ∅ → (𝐴𝑧) ≼ ω))
2322imp 393 . . . . . . . . . . . 12 (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴𝑧) ≼ ω)
24 ctex 8123 . . . . . . . . . . . . . . 15 ((𝐴𝑧) ≼ ω → (𝐴𝑧) ∈ V)
2524adantl 467 . . . . . . . . . . . . . 14 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴𝑧) ∈ V)
26 simpllr 752 . . . . . . . . . . . . . . 15 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → 𝑧𝑦)
27 elssuni 4601 . . . . . . . . . . . . . . 15 (𝑧𝑦𝑧 𝑦)
28 sscon 3893 . . . . . . . . . . . . . . 15 (𝑧 𝑦 → (𝐴 𝑦) ⊆ (𝐴𝑧))
2926, 27, 283syl 18 . . . . . . . . . . . . . 14 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ⊆ (𝐴𝑧))
30 ssdomg 8154 . . . . . . . . . . . . . 14 ((𝐴𝑧) ∈ V → ((𝐴 𝑦) ⊆ (𝐴𝑧) → (𝐴 𝑦) ≼ (𝐴𝑧)))
3125, 29, 30sylc 65 . . . . . . . . . . . . 13 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ (𝐴𝑧))
32 domtr 8161 . . . . . . . . . . . . 13 (((𝐴 𝑦) ≼ (𝐴𝑧) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ ω)
3331, 32sylancom 568 . . . . . . . . . . . 12 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ ω)
3423, 33mpdan 659 . . . . . . . . . . 11 (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴 𝑦) ≼ ω)
3534exp31 406 . . . . . . . . . 10 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (𝑧𝑦 → (¬ 𝑧 = ∅ → (𝐴 𝑦) ≼ ω)))
3635rexlimdv 3177 . . . . . . . . 9 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (∃𝑧𝑦 ¬ 𝑧 = ∅ → (𝐴 𝑦) ≼ ω))
3712, 36syl5bi 232 . . . . . . . 8 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ 𝑦 = ∅ → (𝐴 𝑦) ≼ ω))
3837con1d 141 . . . . . . 7 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ (𝐴 𝑦) ≼ ω → 𝑦 = ∅))
3938orrd 843 . . . . . 6 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅))
40 difeq2 3871 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴 𝑦))
4140breq1d 4794 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴𝑥) ≼ ω ↔ (𝐴 𝑦) ≼ ω))
42 eqeq1 2774 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
4341, 42orbi12d 883 . . . . . . 7 (𝑥 = 𝑦 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅)))
4443elrab 3513 . . . . . 6 ( 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ ( 𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅)))
458, 39, 44sylanbrc 564 . . . . 5 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
4645ax-gen 1869 . . . 4 𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
47 difeq2 3871 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
4847breq1d 4794 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐴𝑥) ≼ ω ↔ (𝐴𝑦) ≼ ω))
49 eqeq1 2774 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5048, 49orbi12d 883 . . . . . . . 8 (𝑥 = 𝑦 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)))
5150elrab 3513 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)))
52 ssinss1 3988 . . . . . . . . . 10 (𝑦𝐴 → (𝑦𝑧) ⊆ 𝐴)
53 vex 3352 . . . . . . . . . . 11 𝑦 ∈ V
5453elpw 4301 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
5553inex1 4930 . . . . . . . . . . 11 (𝑦𝑧) ∈ V
5655elpw 4301 . . . . . . . . . 10 ((𝑦𝑧) ∈ 𝒫 𝐴 ↔ (𝑦𝑧) ⊆ 𝐴)
5752, 54, 563imtr4i 281 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴 → (𝑦𝑧) ∈ 𝒫 𝐴)
5857ad2antrr 697 . . . . . . . 8 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → (𝑦𝑧) ∈ 𝒫 𝐴)
59 difindi 4028 . . . . . . . . . . . 12 (𝐴 ∖ (𝑦𝑧)) = ((𝐴𝑦) ∪ (𝐴𝑧))
60 unctb 9228 . . . . . . . . . . . 12 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → ((𝐴𝑦) ∪ (𝐴𝑧)) ≼ ω)
6159, 60syl5eqbr 4819 . . . . . . . . . . 11 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → (𝐴 ∖ (𝑦𝑧)) ≼ ω)
6261orcd 853 . . . . . . . . . 10 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
63 ineq1 3956 . . . . . . . . . . . 12 (𝑦 = ∅ → (𝑦𝑧) = (∅ ∩ 𝑧))
64 0in 4111 . . . . . . . . . . . 12 (∅ ∩ 𝑧) = ∅
6563, 64syl6eq 2820 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦𝑧) = ∅)
6665olcd 854 . . . . . . . . . 10 (𝑦 = ∅ → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
67 ineq2 3957 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝑦𝑧) = (𝑦 ∩ ∅))
68 in0 4110 . . . . . . . . . . . 12 (𝑦 ∩ ∅) = ∅
6967, 68syl6eq 2820 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑦𝑧) = ∅)
7069olcd 854 . . . . . . . . . 10 (𝑧 = ∅ → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7162, 66, 70ccase2 1024 . . . . . . . . 9 ((((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅) ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7271ad2ant2l 732 . . . . . . . 8 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7358, 72jca 495 . . . . . . 7 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7451, 18, 73syl2anb 577 . . . . . 6 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
75 difeq2 3871 . . . . . . . . 9 (𝑥 = (𝑦𝑧) → (𝐴𝑥) = (𝐴 ∖ (𝑦𝑧)))
7675breq1d 4794 . . . . . . . 8 (𝑥 = (𝑦𝑧) → ((𝐴𝑥) ≼ ω ↔ (𝐴 ∖ (𝑦𝑧)) ≼ ω))
77 eqeq1 2774 . . . . . . . 8 (𝑥 = (𝑦𝑧) → (𝑥 = ∅ ↔ (𝑦𝑧) = ∅))
7876, 77orbi12d 883 . . . . . . 7 (𝑥 = (𝑦𝑧) → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7978elrab 3513 . . . . . 6 ((𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
8074, 79sylibr 224 . . . . 5 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
8180rgen2a 3125 . . . 4 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}
8246, 81pm3.2i 447 . . 3 (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
83 pwexg 4978 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
84 rabexg 4942 . . . 4 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ V)
85 istopg 20919 . . . 4 ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ V → ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})))
8683, 84, 853syl 18 . . 3 (𝐴𝑉 → ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})))
8782, 86mpbiri 248 . 2 (𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top)
88 pwidg 4310 . . . . 5 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
89 omex 8703 . . . . . . . 8 ω ∈ V
90890dom 8245 . . . . . . 7 ∅ ≼ ω
9190orci 845 . . . . . 6 (∅ ≼ ω ∨ 𝐴 = ∅)
9291a1i 11 . . . . 5 (𝐴𝑉 → (∅ ≼ ω ∨ 𝐴 = ∅))
93 difeq2 3871 . . . . . . . . 9 (𝑥 = 𝐴 → (𝐴𝑥) = (𝐴𝐴))
94 difid 4093 . . . . . . . . 9 (𝐴𝐴) = ∅
9593, 94syl6eq 2820 . . . . . . . 8 (𝑥 = 𝐴 → (𝐴𝑥) = ∅)
9695breq1d 4794 . . . . . . 7 (𝑥 = 𝐴 → ((𝐴𝑥) ≼ ω ↔ ∅ ≼ ω))
97 eqeq1 2774 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
9896, 97orbi12d 883 . . . . . 6 (𝑥 = 𝐴 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ (∅ ≼ ω ∨ 𝐴 = ∅)))
9998elrab 3513 . . . . 5 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝐴 ∈ 𝒫 𝐴 ∧ (∅ ≼ ω ∨ 𝐴 = ∅)))
10088, 92, 99sylanbrc 564 . . . 4 (𝐴𝑉𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
101 elssuni 4601 . . . 4 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
102100, 101syl 17 . . 3 (𝐴𝑉𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
1034a1i 11 . . 3 (𝐴𝑉 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴)
104102, 103eqssd 3767 . 2 (𝐴𝑉𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
105 istopon 20936 . 2 ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴) ↔ ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top ∧ 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}))
10687, 104, 105sylanbrc 564 1 (𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826  wal 1628   = wceq 1630  wcel 2144  wral 3060  wrex 3061  {crab 3064  Vcvv 3349  cdif 3718  cun 3719  cin 3720  wss 3721  c0 4061  𝒫 cpw 4295   cuni 4572   class class class wbr 4784  cfv 6031  ωcom 7211  cdom 8106  Topctop 20917  TopOnctopon 20934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-oi 8570  df-card 8964  df-cda 9191  df-top 20918  df-topon 20935
This theorem is referenced by: (None)
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