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Theorem cctop 22922
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 difeq2 4114 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴 𝑦))
21breq1d 5158 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴𝑥) ≼ ω ↔ (𝐴 𝑦) ≼ ω))
3 eqeq1 2732 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
42, 3orbi12d 917 . . . . . 6 (𝑥 = 𝑦 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅)))
5 uniss 4916 . . . . . . . 8 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
6 ssrab2 4075 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴
7 sspwuni 5103 . . . . . . . . 9 ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴)
86, 7mpbi 229 . . . . . . . 8 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴
95, 8sstrdi 3992 . . . . . . 7 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦𝐴)
10 vuniex 7744 . . . . . . . 8 𝑦 ∈ V
1110elpw 4607 . . . . . . 7 ( 𝑦 ∈ 𝒫 𝐴 𝑦𝐴)
129, 11sylibr 233 . . . . . 6 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ 𝒫 𝐴)
13 uni0c 4937 . . . . . . . . . . 11 ( 𝑦 = ∅ ↔ ∀𝑧𝑦 𝑧 = ∅)
1413notbii 320 . . . . . . . . . 10 𝑦 = ∅ ↔ ¬ ∀𝑧𝑦 𝑧 = ∅)
15 rexnal 3097 . . . . . . . . . 10 (∃𝑧𝑦 ¬ 𝑧 = ∅ ↔ ¬ ∀𝑧𝑦 𝑧 = ∅)
1614, 15bitr4i 278 . . . . . . . . 9 𝑦 = ∅ ↔ ∃𝑧𝑦 ¬ 𝑧 = ∅)
17 ssel2 3975 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
18 difeq2 4114 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → (𝐴𝑥) = (𝐴𝑧))
1918breq1d 5158 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → ((𝐴𝑥) ≼ ω ↔ (𝐴𝑧) ≼ ω))
20 eqeq1 2732 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
2119, 20orbi12d 917 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
2221elrab 3682 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
2317, 22sylib 217 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
2423simprd 495 . . . . . . . . . . . . . 14 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))
2524ord 863 . . . . . . . . . . . . 13 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (¬ (𝐴𝑧) ≼ ω → 𝑧 = ∅))
2625con1d 145 . . . . . . . . . . . 12 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (¬ 𝑧 = ∅ → (𝐴𝑧) ≼ ω))
2726imp 406 . . . . . . . . . . 11 (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴𝑧) ≼ ω)
28 ctex 8984 . . . . . . . . . . . . . 14 ((𝐴𝑧) ≼ ω → (𝐴𝑧) ∈ V)
2928adantl 481 . . . . . . . . . . . . 13 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴𝑧) ∈ V)
30 simpllr 775 . . . . . . . . . . . . . 14 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → 𝑧𝑦)
31 elssuni 4940 . . . . . . . . . . . . . 14 (𝑧𝑦𝑧 𝑦)
32 sscon 4137 . . . . . . . . . . . . . 14 (𝑧 𝑦 → (𝐴 𝑦) ⊆ (𝐴𝑧))
3330, 31, 323syl 18 . . . . . . . . . . . . 13 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ⊆ (𝐴𝑧))
34 ssdomg 9021 . . . . . . . . . . . . 13 ((𝐴𝑧) ∈ V → ((𝐴 𝑦) ⊆ (𝐴𝑧) → (𝐴 𝑦) ≼ (𝐴𝑧)))
3529, 33, 34sylc 65 . . . . . . . . . . . 12 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ (𝐴𝑧))
36 domtr 9028 . . . . . . . . . . . 12 (((𝐴 𝑦) ≼ (𝐴𝑧) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ ω)
3735, 36sylancom 587 . . . . . . . . . . 11 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ ω)
3827, 37mpdan 686 . . . . . . . . . 10 (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴 𝑦) ≼ ω)
3938rexlimdva2 3154 . . . . . . . . 9 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (∃𝑧𝑦 ¬ 𝑧 = ∅ → (𝐴 𝑦) ≼ ω))
4016, 39biimtrid 241 . . . . . . . 8 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ 𝑦 = ∅ → (𝐴 𝑦) ≼ ω))
4140con1d 145 . . . . . . 7 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ (𝐴 𝑦) ≼ ω → 𝑦 = ∅))
4241orrd 862 . . . . . 6 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅))
434, 12, 42elrabd 3684 . . . . 5 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
4443ax-gen 1790 . . . 4 𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
45 difeq2 4114 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
4645breq1d 5158 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐴𝑥) ≼ ω ↔ (𝐴𝑦) ≼ ω))
47 eqeq1 2732 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
4846, 47orbi12d 917 . . . . . . . 8 (𝑥 = 𝑦 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)))
4948elrab 3682 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)))
50 ssinss1 4238 . . . . . . . . . 10 (𝑦𝐴 → (𝑦𝑧) ⊆ 𝐴)
51 vex 3475 . . . . . . . . . . 11 𝑦 ∈ V
5251elpw 4607 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
5351inex1 5317 . . . . . . . . . . 11 (𝑦𝑧) ∈ V
5453elpw 4607 . . . . . . . . . 10 ((𝑦𝑧) ∈ 𝒫 𝐴 ↔ (𝑦𝑧) ⊆ 𝐴)
5550, 52, 543imtr4i 292 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴 → (𝑦𝑧) ∈ 𝒫 𝐴)
5655ad2antrr 725 . . . . . . . 8 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → (𝑦𝑧) ∈ 𝒫 𝐴)
57 difindi 4282 . . . . . . . . . . . 12 (𝐴 ∖ (𝑦𝑧)) = ((𝐴𝑦) ∪ (𝐴𝑧))
58 unctb 10229 . . . . . . . . . . . 12 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → ((𝐴𝑦) ∪ (𝐴𝑧)) ≼ ω)
5957, 58eqbrtrid 5183 . . . . . . . . . . 11 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → (𝐴 ∖ (𝑦𝑧)) ≼ ω)
6059orcd 872 . . . . . . . . . 10 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
61 ineq1 4205 . . . . . . . . . . . 12 (𝑦 = ∅ → (𝑦𝑧) = (∅ ∩ 𝑧))
62 0in 4394 . . . . . . . . . . . 12 (∅ ∩ 𝑧) = ∅
6361, 62eqtrdi 2784 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦𝑧) = ∅)
6463olcd 873 . . . . . . . . . 10 (𝑦 = ∅ → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
65 ineq2 4206 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝑦𝑧) = (𝑦 ∩ ∅))
66 in0 4392 . . . . . . . . . . . 12 (𝑦 ∩ ∅) = ∅
6765, 66eqtrdi 2784 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑦𝑧) = ∅)
6867olcd 873 . . . . . . . . . 10 (𝑧 = ∅ → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
6960, 64, 68ccase2 1038 . . . . . . . . 9 ((((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅) ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7069ad2ant2l 745 . . . . . . . 8 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7156, 70jca 511 . . . . . . 7 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7249, 22, 71syl2anb 597 . . . . . 6 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
73 difeq2 4114 . . . . . . . . 9 (𝑥 = (𝑦𝑧) → (𝐴𝑥) = (𝐴 ∖ (𝑦𝑧)))
7473breq1d 5158 . . . . . . . 8 (𝑥 = (𝑦𝑧) → ((𝐴𝑥) ≼ ω ↔ (𝐴 ∖ (𝑦𝑧)) ≼ ω))
75 eqeq1 2732 . . . . . . . 8 (𝑥 = (𝑦𝑧) → (𝑥 = ∅ ↔ (𝑦𝑧) = ∅))
7674, 75orbi12d 917 . . . . . . 7 (𝑥 = (𝑦𝑧) → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7776elrab 3682 . . . . . 6 ((𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7872, 77sylibr 233 . . . . 5 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
7978rgen2 3194 . . . 4 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}
8044, 79pm3.2i 470 . . 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 = ∅)})
81 pwexg 5378 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
82 rabexg 5333 . . . 4 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ V)
83 istopg 22810 . . . 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 = ∅)})))
8481, 82, 833syl 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 = ∅)})))
8580, 84mpbiri 258 . 2 (𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top)
86 difeq2 4114 . . . . . . . 8 (𝑥 = 𝐴 → (𝐴𝑥) = (𝐴𝐴))
87 difid 4371 . . . . . . . 8 (𝐴𝐴) = ∅
8886, 87eqtrdi 2784 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝑥) = ∅)
8988breq1d 5158 . . . . . 6 (𝑥 = 𝐴 → ((𝐴𝑥) ≼ ω ↔ ∅ ≼ ω))
90 eqeq1 2732 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
9189, 90orbi12d 917 . . . . 5 (𝑥 = 𝐴 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ (∅ ≼ ω ∨ 𝐴 = ∅)))
92 pwidg 4623 . . . . 5 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
93 omex 9667 . . . . . . . 8 ω ∈ V
94930dom 9131 . . . . . . 7 ∅ ≼ ω
9594orci 864 . . . . . 6 (∅ ≼ ω ∨ 𝐴 = ∅)
9695a1i 11 . . . . 5 (𝐴𝑉 → (∅ ≼ ω ∨ 𝐴 = ∅))
9791, 92, 96elrabd 3684 . . . 4 (𝐴𝑉𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
98 elssuni 4940 . . . 4 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
9997, 98syl 17 . . 3 (𝐴𝑉𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
1008a1i 11 . . 3 (𝐴𝑉 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴)
10199, 100eqssd 3997 . 2 (𝐴𝑉𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
102 istopon 22827 . 2 ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴) ↔ ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top ∧ 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}))
10385, 101, 102sylanbrc 582 1 (𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846  wal 1532   = wceq 1534  wcel 2099  wral 3058  wrex 3067  {crab 3429  Vcvv 3471  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4323  𝒫 cpw 4603   cuni 4908   class class class wbr 5148  cfv 6548  ωcom 7870  cdom 8962  Topctop 22808  TopOnctopon 22825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9665
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9534  df-dju 9925  df-card 9963  df-top 22809  df-topon 22826
This theorem is referenced by: (None)
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