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Theorem cctop 20620
 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 4394 . . . . . . . 8 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
2 ssrab2 3650 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴
3 sspwuni 4547 . . . . . . . . 9 ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴)
42, 3mpbi 219 . . . . . . . 8 {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴
51, 4syl6ss 3580 . . . . . . 7 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦𝐴)
6 vuniex 6852 . . . . . . . 8 𝑦 ∈ V
76elpw 4114 . . . . . . 7 ( 𝑦 ∈ 𝒫 𝐴 𝑦𝐴)
85, 7sylibr 223 . . . . . 6 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ 𝒫 𝐴)
9 uni0c 4400 . . . . . . . . . . 11 ( 𝑦 = ∅ ↔ ∀𝑧𝑦 𝑧 = ∅)
109notbii 309 . . . . . . . . . 10 𝑦 = ∅ ↔ ¬ ∀𝑧𝑦 𝑧 = ∅)
11 rexnal 2978 . . . . . . . . . 10 (∃𝑧𝑦 ¬ 𝑧 = ∅ ↔ ¬ ∀𝑧𝑦 𝑧 = ∅)
1210, 11bitr4i 266 . . . . . . . . 9 𝑦 = ∅ ↔ ∃𝑧𝑦 ¬ 𝑧 = ∅)
13 ssel2 3563 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
14 difeq2 3684 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑧 → (𝐴𝑥) = (𝐴𝑧))
1514breq1d 4593 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → ((𝐴𝑥) ≼ ω ↔ (𝐴𝑧) ≼ ω))
16 eqeq1 2614 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
1715, 16orbi12d 742 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑧 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
1817elrab 3331 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
1913, 18sylib 207 . . . . . . . . . . . . . . . 16 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)))
2019simprd 478 . . . . . . . . . . . . . . 15 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))
2120ord 391 . . . . . . . . . . . . . 14 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (¬ (𝐴𝑧) ≼ ω → 𝑧 = ∅))
2221con1d 138 . . . . . . . . . . . . 13 ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) → (¬ 𝑧 = ∅ → (𝐴𝑧) ≼ ω))
2322imp 444 . . . . . . . . . . . 12 (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴𝑧) ≼ ω)
24 ctex 7856 . . . . . . . . . . . . . . 15 ((𝐴𝑧) ≼ ω → (𝐴𝑧) ∈ V)
2524adantl 481 . . . . . . . . . . . . . 14 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴𝑧) ∈ V)
26 simpllr 795 . . . . . . . . . . . . . . 15 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → 𝑧𝑦)
27 elssuni 4403 . . . . . . . . . . . . . . 15 (𝑧𝑦𝑧 𝑦)
28 sscon 3706 . . . . . . . . . . . . . . 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 7887 . . . . . . . . . . . . . 14 ((𝐴𝑧) ∈ V → ((𝐴 𝑦) ⊆ (𝐴𝑧) → (𝐴 𝑦) ≼ (𝐴𝑧)))
3125, 29, 30sylc 63 . . . . . . . . . . . . 13 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ (𝐴𝑧))
32 domtr 7895 . . . . . . . . . . . . 13 (((𝐴 𝑦) ≼ (𝐴𝑧) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ ω)
3331, 32sylancom 698 . . . . . . . . . . . 12 ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴𝑧) ≼ ω) → (𝐴 𝑦) ≼ ω)
3423, 33mpdan 699 . . . . . . . . . . 11 (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴 𝑦) ≼ ω)
3534exp31 628 . . . . . . . . . 10 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (𝑧𝑦 → (¬ 𝑧 = ∅ → (𝐴 𝑦) ≼ ω)))
3635rexlimdv 3012 . . . . . . . . 9 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (∃𝑧𝑦 ¬ 𝑧 = ∅ → (𝐴 𝑦) ≼ ω))
3712, 36syl5bi 231 . . . . . . . 8 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ 𝑦 = ∅ → (𝐴 𝑦) ≼ ω))
3837con1d 138 . . . . . . 7 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ (𝐴 𝑦) ≼ ω → 𝑦 = ∅))
3938orrd 392 . . . . . 6 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅))
40 difeq2 3684 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴 𝑦))
4140breq1d 4593 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴𝑥) ≼ ω ↔ (𝐴 𝑦) ≼ ω))
42 eqeq1 2614 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
4341, 42orbi12d 742 . . . . . . 7 (𝑥 = 𝑦 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅)))
4443elrab 3331 . . . . . 6 ( 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ ( 𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴 𝑦) ≼ ω ∨ 𝑦 = ∅)))
458, 39, 44sylanbrc 695 . . . . 5 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
4645ax-gen 1713 . . . 4 𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
47 difeq2 3684 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
4847breq1d 4593 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐴𝑥) ≼ ω ↔ (𝐴𝑦) ≼ ω))
49 eqeq1 2614 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
5048, 49orbi12d 742 . . . . . . . 8 (𝑥 = 𝑦 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)))
5150elrab 3331 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)))
52 ssinss1 3803 . . . . . . . . . 10 (𝑦𝐴 → (𝑦𝑧) ⊆ 𝐴)
53 vex 3176 . . . . . . . . . . 11 𝑦 ∈ V
5453elpw 4114 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
5553inex1 4727 . . . . . . . . . . 11 (𝑦𝑧) ∈ V
5655elpw 4114 . . . . . . . . . 10 ((𝑦𝑧) ∈ 𝒫 𝐴 ↔ (𝑦𝑧) ⊆ 𝐴)
5752, 54, 563imtr4i 280 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴 → (𝑦𝑧) ∈ 𝒫 𝐴)
5857ad2antrr 758 . . . . . . . 8 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → (𝑦𝑧) ∈ 𝒫 𝐴)
59 difindi 3840 . . . . . . . . . . . 12 (𝐴 ∖ (𝑦𝑧)) = ((𝐴𝑦) ∪ (𝐴𝑧))
60 unctb 8910 . . . . . . . . . . . 12 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → ((𝐴𝑦) ∪ (𝐴𝑧)) ≼ ω)
6159, 60syl5eqbr 4618 . . . . . . . . . . 11 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → (𝐴 ∖ (𝑦𝑧)) ≼ ω)
6261orcd 406 . . . . . . . . . 10 (((𝐴𝑦) ≼ ω ∧ (𝐴𝑧) ≼ ω) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
63 ineq1 3769 . . . . . . . . . . . 12 (𝑦 = ∅ → (𝑦𝑧) = (∅ ∩ 𝑧))
64 0in 3921 . . . . . . . . . . . 12 (∅ ∩ 𝑧) = ∅
6563, 64syl6eq 2660 . . . . . . . . . . 11 (𝑦 = ∅ → (𝑦𝑧) = ∅)
6665olcd 407 . . . . . . . . . 10 (𝑦 = ∅ → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
67 ineq2 3770 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝑦𝑧) = (𝑦 ∩ ∅))
68 in0 3920 . . . . . . . . . . . 12 (𝑦 ∩ ∅) = ∅
6967, 68syl6eq 2660 . . . . . . . . . . 11 (𝑧 = ∅ → (𝑦𝑧) = ∅)
7069olcd 407 . . . . . . . . . 10 (𝑧 = ∅ → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7162, 66, 70ccase2 986 . . . . . . . . 9 ((((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅) ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅)) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7271ad2ant2l 778 . . . . . . . 8 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅))
7358, 72jca 553 . . . . . . 7 (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7451, 18, 73syl2anb 495 . . . . . 6 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
75 difeq2 3684 . . . . . . . . 9 (𝑥 = (𝑦𝑧) → (𝐴𝑥) = (𝐴 ∖ (𝑦𝑧)))
7675breq1d 4593 . . . . . . . 8 (𝑥 = (𝑦𝑧) → ((𝐴𝑥) ≼ ω ↔ (𝐴 ∖ (𝑦𝑧)) ≼ ω))
77 eqeq1 2614 . . . . . . . 8 (𝑥 = (𝑦𝑧) → (𝑥 = ∅ ↔ (𝑦𝑧) = ∅))
7876, 77orbi12d 742 . . . . . . 7 (𝑥 = (𝑦𝑧) → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
7978elrab 3331 . . . . . 6 ((𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦𝑧)) ≼ ω ∨ (𝑦𝑧) = ∅)))
8074, 79sylibr 223 . . . . 5 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)}) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
8180rgen2a 2960 . . . 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 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 = ∅)})
83 pwexg 4776 . . . 4 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
84 rabexg 4739 . . . 4 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ V)
85 istopg 20525 . . . 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 247 . 2 (𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top)
88 pwidg 4121 . . . . 5 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
89 omex 8423 . . . . . . . 8 ω ∈ V
90890dom 7975 . . . . . . 7 ∅ ≼ ω
9190orci 404 . . . . . 6 (∅ ≼ ω ∨ 𝐴 = ∅)
9291a1i 11 . . . . 5 (𝐴𝑉 → (∅ ≼ ω ∨ 𝐴 = ∅))
93 difeq2 3684 . . . . . . . . 9 (𝑥 = 𝐴 → (𝐴𝑥) = (𝐴𝐴))
94 difid 3902 . . . . . . . . 9 (𝐴𝐴) = ∅
9593, 94syl6eq 2660 . . . . . . . 8 (𝑥 = 𝐴 → (𝐴𝑥) = ∅)
9695breq1d 4593 . . . . . . 7 (𝑥 = 𝐴 → ((𝐴𝑥) ≼ ω ↔ ∅ ≼ ω))
97 eqeq1 2614 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
9896, 97orbi12d 742 . . . . . 6 (𝑥 = 𝐴 → (((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ (∅ ≼ ω ∨ 𝐴 = ∅)))
9998elrab 3331 . . . . 5 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝐴 ∈ 𝒫 𝐴 ∧ (∅ ≼ ω ∨ 𝐴 = ∅)))
10088, 92, 99sylanbrc 695 . . . 4 (𝐴𝑉𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
101 elssuni 4403 . . . 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 3585 . 2 (𝐴𝑉𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)})
105 istopon 20540 . 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 695 1 (𝐴𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  ‘cfv 5804  ωcom 6957   ≼ cdom 7839  Topctop 20517  TopOnctopon 20518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-cda 8873  df-top 20521  df-topon 20523 This theorem is referenced by: (None)
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