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Theorem cctop 21542
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 4090 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴 𝑦))
21breq1d 5067 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴𝑥) ≼ ω ↔ (𝐴 𝑦) ≼ ω))
3 eqeq1 2822 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
42, 3orbi12d 912 . . . . . 6 (𝑥 = 𝑦 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅)))
5 uniss 4851 . . . . . . . 8 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
6 ssrab2 4053 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴
7 sspwuni 5013 . . . . . . . . 9 ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴)
86, 7mpbi 231 . . . . . . . 8 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴
95, 8sstrdi 3976 . . . . . . 7 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦𝐴)
10 vuniex 7455 . . . . . . . 8 𝑦 ∈ V
1110elpw 4542 . . . . . . 7 ( 𝑦 ∈ 𝒫 𝐴 𝑦𝐴)
129, 11sylibr 235 . . . . . 6 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ 𝒫 𝐴)
13 uni0c 4856 . . . . . . . . . . 11 ( 𝑦 = ∅ ↔ ∀𝑧𝑦 𝑧 = ∅)
1413notbii 321 . . . . . . . . . 10 𝑦 = ∅ ↔ ¬ ∀𝑧𝑦 𝑧 = ∅)
15 rexnal 3235 . . . . . . . . . 10 (∃𝑧𝑦 ¬ 𝑧 = ∅ ↔ ¬ ∀𝑧𝑦 𝑧 = ∅)
1614, 15bitr4i 279 . . . . . . . . 9 𝑦 = ∅ ↔ ∃𝑧𝑦 ¬ 𝑧 = ∅)
17 ssel2 3959 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
18 difeq2 4090 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → (𝐴𝑥) = (𝐴𝑧))
1918breq1d 5067 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → ((𝐴𝑥) ≼ ω ↔ (𝐴𝑧) ≼ ω))
20 eqeq1 2822 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
2119, 20orbi12d 912 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
2221elrab 3677 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
2317, 22sylib 219 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
2423simprd 496 . . . . . . . . . . . . . 14 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))
2524ord 858 . . . . . . . . . . . . 13 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (¬ (𝐴𝑧) ≼ ω → 𝑧 = ∅))
2625con1d 147 . . . . . . . . . . . 12 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (¬ 𝑧 = ∅ → (𝐴𝑧) ≼ ω))
2726imp 407 . . . . . . . . . . 11 (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴𝑧) ≼ ω)
28 ctex 8512 . . . . . . . . . . . . . 14 ((𝐴𝑧) ≼ ω → (𝐴𝑧) ∈ V)
2928adantl 482 . . . . . . . . . . . . 13 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴𝑧) ∈ V)
30 simpllr 772 . . . . . . . . . . . . . 14 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → 𝑧𝑦)
31 elssuni 4859 . . . . . . . . . . . . . 14 (𝑧𝑦𝑧 𝑦)
32 sscon 4112 . . . . . . . . . . . . . 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 8543 . . . . . . . . . . . . 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 8550 . . . . . . . . . . . 12 (((𝐴 𝑦) ≼ (𝐴𝑧) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ ω)
3735, 36sylancom 588 . . . . . . . . . . 11 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ ω)
3827, 37mpdan 683 . . . . . . . . . 10 (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴 𝑦) ≼ ω)
3938rexlimdva2 3284 . . . . . . . . 9 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (∃𝑧𝑦 ¬ 𝑧 = ∅ → (𝐴 𝑦) ≼ ω))
4016, 39syl5bi 243 . . . . . . . 8 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ 𝑦 = ∅ → (𝐴 𝑦) ≼ ω))
4140con1d 147 . . . . . . 7 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ (𝐴 𝑦) ≼ ω → 𝑦 = ∅))
4241orrd 857 . . . . . 6 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅))
434, 12, 42elrabd 3679 . . . . 5 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
4443ax-gen 1787 . . . 4 𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
45 difeq2 4090 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
4645breq1d 5067 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐴𝑥) ≼ ω ↔ (𝐴𝑦) ≼ ω))
47 eqeq1 2822 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
4846, 47orbi12d 912 . . . . . . . 8 (𝑥 = 𝑦 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)))
4948elrab 3677 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)))
50 ssinss1 4211 . . . . . . . . . 10 (𝑦𝐴 → (𝑦𝑧) ⊆ 𝐴)
51 vex 3495 . . . . . . . . . . 11 𝑦 ∈ V
5251elpw 4542 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
5351inex1 5212 . . . . . . . . . . 11 (𝑦𝑧) ∈ V
5453elpw 4542 . . . . . . . . . 10 ((𝑦𝑧) ∈ 𝒫 𝐴 ↔ (𝑦𝑧) ⊆ 𝐴)
5550, 52, 543imtr4i 293 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴 → (𝑦𝑧) ∈ 𝒫 𝐴)
5655ad2antrr 722 . . . . . . . 8 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → (𝑦𝑧) ∈ 𝒫 𝐴)
57 difindi 4255 . . . . . . . . . . . 12 (𝐴 ∖ (𝑦𝑧)) = ((𝐴𝑦) ∪ (𝐴𝑧))
58 unctb 9615 . . . . . . . . . . . 12 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → ((𝐴𝑦) ∪ (𝐴𝑧)) ≼ ω)
5957, 58eqbrtrid 5092 . . . . . . . . . . 11 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → (𝐴 ∖ (𝑦𝑧)) ≼ ω)
6059orcd 869 . . . . . . . . . 10 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
61 ineq1 4178 . . . . . . . . . . . 12 (𝑦 = ∅ → (𝑦𝑧) = (∅ ∩ 𝑧))
62 0in 4344 . . . . . . . . . . . 12 (∅ ∩ 𝑧) = ∅
6361, 62syl6eq 2869 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦𝑧) = ∅)
6463olcd 870 . . . . . . . . . 10 (𝑦 = ∅ → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
65 ineq2 4180 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝑦𝑧) = (𝑦 ∩ ∅))
66 in0 4342 . . . . . . . . . . . 12 (𝑦 ∩ ∅) = ∅
6765, 66syl6eq 2869 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑦𝑧) = ∅)
6867olcd 870 . . . . . . . . . 10 (𝑧 = ∅ → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
6960, 64, 68ccase2 1031 . . . . . . . . 9 ((((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅) ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7069ad2ant2l 742 . . . . . . . 8 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7156, 70jca 512 . . . . . . 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 4090 . . . . . . . . 9 (𝑥 = (𝑦𝑧) → (𝐴𝑥) = (𝐴 ∖ (𝑦𝑧)))
7473breq1d 5067 . . . . . . . 8 (𝑥 = (𝑦𝑧) → ((𝐴𝑥) ≼ ω ↔ (𝐴 ∖ (𝑦𝑧)) ≼ ω))
75 eqeq1 2822 . . . . . . . 8 (𝑥 = (𝑦𝑧) → (𝑥 = ∅ ↔ (𝑦𝑧) = ∅))
7674, 75orbi12d 912 . . . . . . 7 (𝑥 = (𝑦𝑧) → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7776elrab 3677 . . . . . 6 ((𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7872, 77sylibr 235 . . . . 5 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
7978rgen2 3200 . . . 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 471 . . 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 5270 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
82 rabexg 5225 . . . 4 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ V)
83 istopg 21431 . . . 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 259 . 2 (𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top)
86 difeq2 4090 . . . . . . . 8 (𝑥 = 𝐴 → (𝐴𝑥) = (𝐴𝐴))
87 difid 4327 . . . . . . . 8 (𝐴𝐴) = ∅
8886, 87syl6eq 2869 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝑥) = ∅)
8988breq1d 5067 . . . . . 6 (𝑥 = 𝐴 → ((𝐴𝑥) ≼ ω ↔ ∅ ≼ ω))
90 eqeq1 2822 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
9189, 90orbi12d 912 . . . . 5 (𝑥 = 𝐴 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ (∅ ≼ ω ∨ 𝐴 = ∅)))
92 pwidg 4552 . . . . 5 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
93 omex 9094 . . . . . . . 8 ω ∈ V
94930dom 8635 . . . . . . 7 ∅ ≼ ω
9594orci 859 . . . . . 6 (∅ ≼ ω ∨ 𝐴 = ∅)
9695a1i 11 . . . . 5 (𝐴𝑉 → (∅ ≼ ω ∨ 𝐴 = ∅))
9791, 92, 96elrabd 3679 . . . 4 (𝐴𝑉𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
98 elssuni 4859 . . . 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 3981 . 2 (𝐴𝑉𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
102 istopon 21448 . 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 583 1 (𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  wal 1526   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  cdif 3930  cun 3931  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  cfv 6348  ωcom 7569  cdom 8495  Topctop 21429  TopOnctopon 21446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-dju 9318  df-card 9356  df-top 21430  df-topon 21447
This theorem is referenced by: (None)
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