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Theorem ppttop 20621
Description: The particular point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
ppttop ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ (TopOn‘𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑃   𝑥,𝑉

Proof of Theorem ppttop
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab 3643 . . . . 5 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅)))
2 simprl 790 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅))) → 𝑦 ⊆ 𝒫 𝐴)
3 sspwuni 4547 . . . . . . . . 9 (𝑦 ⊆ 𝒫 𝐴 𝑦𝐴)
42, 3sylib 207 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅))) → 𝑦𝐴)
5 vuniex 6852 . . . . . . . . 9 𝑦 ∈ V
65elpw 4114 . . . . . . . 8 ( 𝑦 ∈ 𝒫 𝐴 𝑦𝐴)
74, 6sylibr 223 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅))) → 𝑦 ∈ 𝒫 𝐴)
8 neq0 3889 . . . . . . . . . 10 𝑦 = ∅ ↔ ∃𝑧 𝑧 𝑦)
9 eluni2 4376 . . . . . . . . . . . 12 (𝑧 𝑦 ↔ ∃𝑥𝑦 𝑧𝑥)
10 r19.29 3054 . . . . . . . . . . . . . . 15 ((∀𝑥𝑦 (𝑃𝑥𝑥 = ∅) ∧ ∃𝑥𝑦 𝑧𝑥) → ∃𝑥𝑦 ((𝑃𝑥𝑥 = ∅) ∧ 𝑧𝑥))
11 n0i 3879 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑥 → ¬ 𝑥 = ∅)
1211adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝑃𝑥𝑥 = ∅) ∧ 𝑧𝑥) → ¬ 𝑥 = ∅)
13 simpl 472 . . . . . . . . . . . . . . . . . . . 20 (((𝑃𝑥𝑥 = ∅) ∧ 𝑧𝑥) → (𝑃𝑥𝑥 = ∅))
1413ord 391 . . . . . . . . . . . . . . . . . . 19 (((𝑃𝑥𝑥 = ∅) ∧ 𝑧𝑥) → (¬ 𝑃𝑥𝑥 = ∅))
1512, 14mt3d 139 . . . . . . . . . . . . . . . . . 18 (((𝑃𝑥𝑥 = ∅) ∧ 𝑧𝑥) → 𝑃𝑥)
1615adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = ∅) ∧ 𝑧𝑥)) → 𝑃𝑥)
17 simpl 472 . . . . . . . . . . . . . . . . 17 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = ∅) ∧ 𝑧𝑥)) → 𝑥𝑦)
18 elunii 4377 . . . . . . . . . . . . . . . . 17 ((𝑃𝑥𝑥𝑦) → 𝑃 𝑦)
1916, 17, 18syl2anc 691 . . . . . . . . . . . . . . . 16 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = ∅) ∧ 𝑧𝑥)) → 𝑃 𝑦)
2019rexlimiva 3010 . . . . . . . . . . . . . . 15 (∃𝑥𝑦 ((𝑃𝑥𝑥 = ∅) ∧ 𝑧𝑥) → 𝑃 𝑦)
2110, 20syl 17 . . . . . . . . . . . . . 14 ((∀𝑥𝑦 (𝑃𝑥𝑥 = ∅) ∧ ∃𝑥𝑦 𝑧𝑥) → 𝑃 𝑦)
2221ex 449 . . . . . . . . . . . . 13 (∀𝑥𝑦 (𝑃𝑥𝑥 = ∅) → (∃𝑥𝑦 𝑧𝑥𝑃 𝑦))
2322ad2antll 761 . . . . . . . . . . . 12 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅))) → (∃𝑥𝑦 𝑧𝑥𝑃 𝑦))
249, 23syl5bi 231 . . . . . . . . . . 11 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅))) → (𝑧 𝑦𝑃 𝑦))
2524exlimdv 1848 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅))) → (∃𝑧 𝑧 𝑦𝑃 𝑦))
268, 25syl5bi 231 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅))) → (¬ 𝑦 = ∅ → 𝑃 𝑦))
2726con1d 138 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅))) → (¬ 𝑃 𝑦 𝑦 = ∅))
2827orrd 392 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅))) → (𝑃 𝑦 𝑦 = ∅))
29 eleq2 2677 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑃𝑥𝑃 𝑦))
30 eqeq1 2614 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
3129, 30orbi12d 742 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑃𝑥𝑥 = ∅) ↔ (𝑃 𝑦 𝑦 = ∅)))
3231elrab 3331 . . . . . . 7 ( 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ↔ ( 𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 𝑦 𝑦 = ∅)))
337, 28, 32sylanbrc 695 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅))) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)})
3433ex 449 . . . . 5 ((𝐴𝑉𝑃𝐴) → ((𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = ∅)) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}))
351, 34syl5bi 231 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}))
3635alrimiv 1842 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}))
37 eleq2 2677 . . . . . . . 8 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
38 eqeq1 2614 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
3937, 38orbi12d 742 . . . . . . 7 (𝑥 = 𝑦 → ((𝑃𝑥𝑥 = ∅) ↔ (𝑃𝑦𝑦 = ∅)))
4039elrab 3331 . . . . . 6 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)))
41 eleq2 2677 . . . . . . . 8 (𝑥 = 𝑧 → (𝑃𝑥𝑃𝑧))
42 eqeq1 2614 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
4341, 42orbi12d 742 . . . . . . 7 (𝑥 = 𝑧 → ((𝑃𝑥𝑥 = ∅) ↔ (𝑃𝑧𝑧 = ∅)))
4443elrab 3331 . . . . . 6 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))
4540, 44anbi12i 729 . . . . 5 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}) ↔ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))))
46 inss1 3795 . . . . . . . . 9 (𝑦𝑧) ⊆ 𝑦
47 simprll 798 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → 𝑦 ∈ 𝒫 𝐴)
4847elpwid 4118 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → 𝑦𝐴)
4946, 48syl5ss 3579 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → (𝑦𝑧) ⊆ 𝐴)
50 vex 3176 . . . . . . . . . 10 𝑦 ∈ V
5150inex1 4727 . . . . . . . . 9 (𝑦𝑧) ∈ V
5251elpw 4114 . . . . . . . 8 ((𝑦𝑧) ∈ 𝒫 𝐴 ↔ (𝑦𝑧) ⊆ 𝐴)
5349, 52sylibr 223 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → (𝑦𝑧) ∈ 𝒫 𝐴)
54 ianor 508 . . . . . . . . . . 11 (¬ (𝑃𝑦𝑃𝑧) ↔ (¬ 𝑃𝑦 ∨ ¬ 𝑃𝑧))
55 elin 3758 . . . . . . . . . . 11 (𝑃 ∈ (𝑦𝑧) ↔ (𝑃𝑦𝑃𝑧))
5654, 55xchnxbir 322 . . . . . . . . . 10 𝑃 ∈ (𝑦𝑧) ↔ (¬ 𝑃𝑦 ∨ ¬ 𝑃𝑧))
57 simprlr 799 . . . . . . . . . . . 12 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → (𝑃𝑦𝑦 = ∅))
5857ord 391 . . . . . . . . . . 11 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → (¬ 𝑃𝑦𝑦 = ∅))
59 simprrr 801 . . . . . . . . . . . 12 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → (𝑃𝑧𝑧 = ∅))
6059ord 391 . . . . . . . . . . 11 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → (¬ 𝑃𝑧𝑧 = ∅))
6158, 60orim12d 879 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → ((¬ 𝑃𝑦 ∨ ¬ 𝑃𝑧) → (𝑦 = ∅ ∨ 𝑧 = ∅)))
6256, 61syl5bi 231 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → (¬ 𝑃 ∈ (𝑦𝑧) → (𝑦 = ∅ ∨ 𝑧 = ∅)))
63 inss 3804 . . . . . . . . . 10 ((𝑦 ⊆ ∅ ∨ 𝑧 ⊆ ∅) → (𝑦𝑧) ⊆ ∅)
64 ss0b 3925 . . . . . . . . . . 11 (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
65 ss0b 3925 . . . . . . . . . . 11 (𝑧 ⊆ ∅ ↔ 𝑧 = ∅)
6664, 65orbi12i 542 . . . . . . . . . 10 ((𝑦 ⊆ ∅ ∨ 𝑧 ⊆ ∅) ↔ (𝑦 = ∅ ∨ 𝑧 = ∅))
67 ss0b 3925 . . . . . . . . . 10 ((𝑦𝑧) ⊆ ∅ ↔ (𝑦𝑧) = ∅)
6863, 66, 673imtr3i 279 . . . . . . . . 9 ((𝑦 = ∅ ∨ 𝑧 = ∅) → (𝑦𝑧) = ∅)
6962, 68syl6 34 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → (¬ 𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = ∅))
7069orrd 392 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → (𝑃 ∈ (𝑦𝑧) ∨ (𝑦𝑧) = ∅))
71 eleq2 2677 . . . . . . . . 9 (𝑥 = (𝑦𝑧) → (𝑃𝑥𝑃 ∈ (𝑦𝑧)))
72 eqeq1 2614 . . . . . . . . 9 (𝑥 = (𝑦𝑧) → (𝑥 = ∅ ↔ (𝑦𝑧) = ∅))
7371, 72orbi12d 742 . . . . . . . 8 (𝑥 = (𝑦𝑧) → ((𝑃𝑥𝑥 = ∅) ↔ (𝑃 ∈ (𝑦𝑧) ∨ (𝑦𝑧) = ∅)))
7473elrab 3331 . . . . . . 7 ((𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ↔ ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) ∨ (𝑦𝑧) = ∅)))
7553, 70, 74sylanbrc 695 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)})
7675ex 449 . . . . 5 ((𝐴𝑉𝑃𝐴) → (((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}))
7745, 76syl5bi 231 . . . 4 ((𝐴𝑉𝑃𝐴) → ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}))
7877ralrimivv 2953 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)})
79 pwexg 4776 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
8079adantr 480 . . . . 5 ((𝐴𝑉𝑃𝐴) → 𝒫 𝐴 ∈ V)
81 rabexg 4739 . . . . 5 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ V)
8280, 81syl 17 . . . 4 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ V)
83 istopg 20525 . . . 4 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ V → ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)})))
8482, 83syl 17 . . 3 ((𝐴𝑉𝑃𝐴) → ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)})))
8536, 78, 84mpbir2and 959 . 2 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ Top)
86 pwidg 4121 . . . . . 6 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
8786adantr 480 . . . . 5 ((𝐴𝑉𝑃𝐴) → 𝐴 ∈ 𝒫 𝐴)
88 simpr 476 . . . . . 6 ((𝐴𝑉𝑃𝐴) → 𝑃𝐴)
8988orcd 406 . . . . 5 ((𝐴𝑉𝑃𝐴) → (𝑃𝐴𝐴 = ∅))
90 eleq2 2677 . . . . . . 7 (𝑥 = 𝐴 → (𝑃𝑥𝑃𝐴))
91 eqeq1 2614 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
9290, 91orbi12d 742 . . . . . 6 (𝑥 = 𝐴 → ((𝑃𝑥𝑥 = ∅) ↔ (𝑃𝐴𝐴 = ∅)))
9392elrab 3331 . . . . 5 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ↔ (𝐴 ∈ 𝒫 𝐴 ∧ (𝑃𝐴𝐴 = ∅)))
9487, 89, 93sylanbrc 695 . . . 4 ((𝐴𝑉𝑃𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)})
95 elssuni 4403 . . . 4 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} → 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)})
9694, 95syl 17 . . 3 ((𝐴𝑉𝑃𝐴) → 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)})
97 ssrab2 3650 . . . . 5 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ⊆ 𝒫 𝐴
98 sspwuni 4547 . . . . 5 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ⊆ 𝒫 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ⊆ 𝐴)
9997, 98mpbi 219 . . . 4 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ⊆ 𝐴
10099a1i 11 . . 3 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ⊆ 𝐴)
10196, 100eqssd 3585 . 2 ((𝐴𝑉𝑃𝐴) → 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)})
102 istopon 20540 . 2 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ (TopOn‘𝐴) ↔ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ Top ∧ 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}))
10385, 101, 102sylanbrc 695 1 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383  wal 1473   = wceq 1475  wex 1695  wcel 1977  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  cin 3539  wss 3540  c0 3874  𝒫 cpw 4108   cuni 4372  cfv 5804  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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-topon 20523
This theorem is referenced by:  pptbas  20622
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