MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cctop Structured version   Visualization version   GIF version

Theorem cctop 21308
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 3979 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴 𝑦))
21breq1d 4933 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴𝑥) ≼ ω ↔ (𝐴 𝑦) ≼ ω))
3 eqeq1 2776 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
42, 3orbi12d 902 . . . . . 6 (𝑥 = 𝑦 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅)))
5 uniss 4727 . . . . . . . 8 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
6 ssrab2 3942 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴
7 sspwuni 4882 . . . . . . . . 9 ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴)
86, 7mpbi 222 . . . . . . . 8 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴
95, 8syl6ss 3866 . . . . . . 7 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦𝐴)
10 vuniex 7278 . . . . . . . 8 𝑦 ∈ V
1110elpw 4422 . . . . . . 7 ( 𝑦 ∈ 𝒫 𝐴 𝑦𝐴)
129, 11sylibr 226 . . . . . 6 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ 𝒫 𝐴)
13 uni0c 4732 . . . . . . . . . . 11 ( 𝑦 = ∅ ↔ ∀𝑧𝑦 𝑧 = ∅)
1413notbii 312 . . . . . . . . . 10 𝑦 = ∅ ↔ ¬ ∀𝑧𝑦 𝑧 = ∅)
15 rexnal 3179 . . . . . . . . . 10 (∃𝑧𝑦 ¬ 𝑧 = ∅ ↔ ¬ ∀𝑧𝑦 𝑧 = ∅)
1614, 15bitr4i 270 . . . . . . . . 9 𝑦 = ∅ ↔ ∃𝑧𝑦 ¬ 𝑧 = ∅)
17 ssel2 3849 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
18 difeq2 3979 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → (𝐴𝑥) = (𝐴𝑧))
1918breq1d 4933 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → ((𝐴𝑥) ≼ ω ↔ (𝐴𝑧) ≼ ω))
20 eqeq1 2776 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
2119, 20orbi12d 902 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑧 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
2221elrab 3589 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
2317, 22sylib 210 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
2423simprd 488 . . . . . . . . . . . . . 14 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))
2524ord 850 . . . . . . . . . . . . 13 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (¬ (𝐴𝑧) ≼ ω → 𝑧 = ∅))
2625con1d 142 . . . . . . . . . . . 12 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (¬ 𝑧 = ∅ → (𝐴𝑧) ≼ ω))
2726imp 398 . . . . . . . . . . 11 (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴𝑧) ≼ ω)
28 ctex 8313 . . . . . . . . . . . . . 14 ((𝐴𝑧) ≼ ω → (𝐴𝑧) ∈ V)
2928adantl 474 . . . . . . . . . . . . 13 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴𝑧) ∈ V)
30 simpllr 763 . . . . . . . . . . . . . 14 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → 𝑧𝑦)
31 elssuni 4735 . . . . . . . . . . . . . 14 (𝑧𝑦𝑧 𝑦)
32 sscon 4001 . . . . . . . . . . . . . 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 8344 . . . . . . . . . . . . 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 8351 . . . . . . . . . . . 12 (((𝐴 𝑦) ≼ (𝐴𝑧) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ ω)
3735, 36sylancom 579 . . . . . . . . . . 11 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ ω)
3827, 37mpdan 674 . . . . . . . . . 10 (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴 𝑦) ≼ ω)
3938rexlimdva2 3226 . . . . . . . . 9 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (∃𝑧𝑦 ¬ 𝑧 = ∅ → (𝐴 𝑦) ≼ ω))
4016, 39syl5bi 234 . . . . . . . 8 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ 𝑦 = ∅ → (𝐴 𝑦) ≼ ω))
4140con1d 142 . . . . . . 7 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ (𝐴 𝑦) ≼ ω → 𝑦 = ∅))
4241orrd 849 . . . . . 6 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅))
434, 12, 42elrabd 3592 . . . . 5 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
4443ax-gen 1758 . . . 4 𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
45 difeq2 3979 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
4645breq1d 4933 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐴𝑥) ≼ ω ↔ (𝐴𝑦) ≼ ω))
47 eqeq1 2776 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
4846, 47orbi12d 902 . . . . . . . 8 (𝑥 = 𝑦 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)))
4948elrab 3589 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)))
50 ssinss1 4096 . . . . . . . . . 10 (𝑦𝐴 → (𝑦𝑧) ⊆ 𝐴)
51 vex 3412 . . . . . . . . . . 11 𝑦 ∈ V
5251elpw 4422 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
5351inex1 5072 . . . . . . . . . . 11 (𝑦𝑧) ∈ V
5453elpw 4422 . . . . . . . . . 10 ((𝑦𝑧) ∈ 𝒫 𝐴 ↔ (𝑦𝑧) ⊆ 𝐴)
5550, 52, 543imtr4i 284 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴 → (𝑦𝑧) ∈ 𝒫 𝐴)
5655ad2antrr 713 . . . . . . . 8 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → (𝑦𝑧) ∈ 𝒫 𝐴)
57 difindi 4140 . . . . . . . . . . . 12 (𝐴 ∖ (𝑦𝑧)) = ((𝐴𝑦) ∪ (𝐴𝑧))
58 unctb 9417 . . . . . . . . . . . 12 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → ((𝐴𝑦) ∪ (𝐴𝑧)) ≼ ω)
5957, 58syl5eqbr 4958 . . . . . . . . . . 11 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → (𝐴 ∖ (𝑦𝑧)) ≼ ω)
6059orcd 859 . . . . . . . . . 10 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
61 ineq1 4064 . . . . . . . . . . . 12 (𝑦 = ∅ → (𝑦𝑧) = (∅ ∩ 𝑧))
62 0in 4227 . . . . . . . . . . . 12 (∅ ∩ 𝑧) = ∅
6361, 62syl6eq 2824 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦𝑧) = ∅)
6463olcd 860 . . . . . . . . . 10 (𝑦 = ∅ → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
65 ineq2 4065 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝑦𝑧) = (𝑦 ∩ ∅))
66 in0 4226 . . . . . . . . . . . 12 (𝑦 ∩ ∅) = ∅
6765, 66syl6eq 2824 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑦𝑧) = ∅)
6867olcd 860 . . . . . . . . . 10 (𝑧 = ∅ → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
6960, 64, 68ccase2 1020 . . . . . . . . 9 ((((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅) ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7069ad2ant2l 733 . . . . . . . 8 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7156, 70jca 504 . . . . . . 7 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7249, 22, 71syl2anb 588 . . . . . 6 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
73 difeq2 3979 . . . . . . . . 9 (𝑥 = (𝑦𝑧) → (𝐴𝑥) = (𝐴 ∖ (𝑦𝑧)))
7473breq1d 4933 . . . . . . . 8 (𝑥 = (𝑦𝑧) → ((𝐴𝑥) ≼ ω ↔ (𝐴 ∖ (𝑦𝑧)) ≼ ω))
75 eqeq1 2776 . . . . . . . 8 (𝑥 = (𝑦𝑧) → (𝑥 = ∅ ↔ (𝑦𝑧) = ∅))
7674, 75orbi12d 902 . . . . . . 7 (𝑥 = (𝑦𝑧) → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7776elrab 3589 . . . . . 6 ((𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7872, 77sylibr 226 . . . . 5 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
7978rgen2a 3170 . . . 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 463 . . 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 5126 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
82 rabexg 5084 . . . 4 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ V)
83 istopg 21197 . . . 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 250 . 2 (𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top)
86 difeq2 3979 . . . . . . . 8 (𝑥 = 𝐴 → (𝐴𝑥) = (𝐴𝐴))
87 difid 4211 . . . . . . . 8 (𝐴𝐴) = ∅
8886, 87syl6eq 2824 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝑥) = ∅)
8988breq1d 4933 . . . . . 6 (𝑥 = 𝐴 → ((𝐴𝑥) ≼ ω ↔ ∅ ≼ ω))
90 eqeq1 2776 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
9189, 90orbi12d 902 . . . . 5 (𝑥 = 𝐴 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ (∅ ≼ ω ∨ 𝐴 = ∅)))
92 pwidg 4431 . . . . 5 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
93 omex 8892 . . . . . . . 8 ω ∈ V
94930dom 8435 . . . . . . 7 ∅ ≼ ω
9594orci 851 . . . . . 6 (∅ ≼ ω ∨ 𝐴 = ∅)
9695a1i 11 . . . . 5 (𝐴𝑉 → (∅ ≼ ω ∨ 𝐴 = ∅))
9791, 92, 96elrabd 3592 . . . 4 (𝐴𝑉𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
98 elssuni 4735 . . . 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 3871 . 2 (𝐴𝑉𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
102 istopon 21214 . 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 575 1 (𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 833  wal 1505   = wceq 1507  wcel 2048  wral 3082  wrex 3083  {crab 3086  Vcvv 3409  cdif 3822  cun 3823  cin 3824  wss 3825  c0 4173  𝒫 cpw 4416   cuni 4706   class class class wbr 4923  cfv 6182  ωcom 7390  cdom 8296  Topctop 21195  TopOnctopon 21212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-inf2 8890
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-2o 7898  df-oadd 7901  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-oi 8761  df-dju 9116  df-card 9154  df-top 21196  df-topon 21213
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator