Theorem ppttop 21025

Description: The particular point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
ppttop	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ (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.

Step	Hyp	Ref	Expression
1		ssrab 3877	. . . . 5 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)))
2		simprl 778	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → 𝑦 ⊆ 𝒫 𝐴)
3		sspwuni 4803	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
4	2, 3	sylib 209	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → ∪ 𝑦 ⊆ 𝐴)
5		vuniex 7184	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
6	5	elpw 4357	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
7	4, 6	sylibr 225	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → ∪ 𝑦 ∈ 𝒫 𝐴)
8		neq0 4131	. . . . . . . . . 10 ⊢ (¬ ∪ 𝑦 = ∅ ↔ ∃𝑧 𝑧 ∈ ∪ 𝑦)
9		eluni2 4634	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥)
10		r19.29 3260	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥) → ∃𝑥 ∈ 𝑦 ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥))
11		n0i 4121	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝑥 → ¬ 𝑥 = ∅)
12	11	adantl 469	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → ¬ 𝑥 = ∅)
13		simpl 470	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))
14	13	ord 882	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → (¬ 𝑃 ∈ 𝑥 → 𝑥 = ∅))
15	12, 14	mt3d 142	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → 𝑃 ∈ 𝑥)
16	15	adantl 469	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥)) → 𝑃 ∈ 𝑥)
17		simpl 470	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥)) → 𝑥 ∈ 𝑦)
18		elunii 4635	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) → 𝑃 ∈ ∪ 𝑦)
19	16, 17, 18	syl2anc 575	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥)) → 𝑃 ∈ ∪ 𝑦)
20	19	rexlimiva 3216	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ 𝑦 ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → 𝑃 ∈ ∪ 𝑦)
21	10, 20	syl 17	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥) → 𝑃 ∈ ∪ 𝑦)
22	21	ex 399	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) → (∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥 → 𝑃 ∈ ∪ 𝑦))
23	22	ad2antll 711	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥 → 𝑃 ∈ ∪ 𝑦))
24	9, 23	syl5bi 233	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (𝑧 ∈ ∪ 𝑦 → 𝑃 ∈ ∪ 𝑦))
25	24	exlimdv 2024	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (∃𝑧 𝑧 ∈ ∪ 𝑦 → 𝑃 ∈ ∪ 𝑦))
26	8, 25	syl5bi 233	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (¬ ∪ 𝑦 = ∅ → 𝑃 ∈ ∪ 𝑦))
27	26	con1d 141	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (¬ 𝑃 ∈ ∪ 𝑦 → ∪ 𝑦 = ∅))
28	27	orrd 881	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (𝑃 ∈ ∪ 𝑦 ∨ ∪ 𝑦 = ∅))
29		eleq2 2874	. . . . . . . . 9 ⊢ (𝑥 = ∪ 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ ∪ 𝑦))
30		eqeq1 2810	. . . . . . . . 9 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 = ∅ ↔ ∪ 𝑦 = ∅))
31	29, 30	orbi12d 933	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ ∪ 𝑦 ∨ ∪ 𝑦 = ∅)))
32	31	elrab 3559	. . . . . . 7 ⊢ (∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (∪ 𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ ∪ 𝑦 ∨ ∪ 𝑦 = ∅)))
33	7, 28, 32	sylanbrc 574	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
34	33	ex 399	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ((𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
35	1, 34	syl5bi 233	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
36	35	alrimiv 2018	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
37		eleq2 2874	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
38		eqeq1 2810	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
39	37, 38	orbi12d 933	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)))
40	39	elrab 3559	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)))
41		eleq2 2874	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧))
42		eqeq1 2810	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
43	41, 42	orbi12d 933	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
44	43	elrab 3559	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
45	40, 44	anbi12i 614	. . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}) ↔ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))))
46		inss1 4029	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ⊆ 𝑦
47		simprll 788	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → 𝑦 ∈ 𝒫 𝐴)
48	47	elpwid 4363	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → 𝑦 ⊆ 𝐴)
49	46, 48	syl5ss 3809	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑦 ∩ 𝑧) ⊆ 𝐴)
50		vex 3394	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
51	50	inex1 4994	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ∈ V
52	51	elpw 4357	. . . . . . . 8 ⊢ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ↔ (𝑦 ∩ 𝑧) ⊆ 𝐴)
53	49, 52	sylibr 225	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑦 ∩ 𝑧) ∈ 𝒫 𝐴)
54		ianor 995	. . . . . . . . . . 11 ⊢ (¬ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧) ↔ (¬ 𝑃 ∈ 𝑦 ∨ ¬ 𝑃 ∈ 𝑧))
55		elin 3995	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (𝑦 ∩ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧))
56	54, 55	xchnxbir 324	. . . . . . . . . 10 ⊢ (¬ 𝑃 ∈ (𝑦 ∩ 𝑧) ↔ (¬ 𝑃 ∈ 𝑦 ∨ ¬ 𝑃 ∈ 𝑧))
57		simprlr 789	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅))
58	57	ord 882	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ 𝑦 → 𝑦 = ∅))
59		simprrr 791	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))
60	59	ord 882	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ 𝑧 → 𝑧 = ∅))
61	58, 60	orim12d 978	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → ((¬ 𝑃 ∈ 𝑦 ∨ ¬ 𝑃 ∈ 𝑧) → (𝑦 = ∅ ∨ 𝑧 = ∅)))
62	56, 61	syl5bi 233	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 = ∅ ∨ 𝑧 = ∅)))
63		inss 4039	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∨ 𝑧 ⊆ ∅) → (𝑦 ∩ 𝑧) ⊆ ∅)
64		ss0b 4171	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
65		ss0b 4171	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ ∅ ↔ 𝑧 = ∅)
66	64, 65	orbi12i 929	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∨ 𝑧 ⊆ ∅) ↔ (𝑦 = ∅ ∨ 𝑧 = ∅))
67		ss0b 4171	. . . . . . . . . 10 ⊢ ((𝑦 ∩ 𝑧) ⊆ ∅ ↔ (𝑦 ∩ 𝑧) = ∅)
68	63, 66, 67	3imtr3i 282	. . . . . . . . 9 ⊢ ((𝑦 = ∅ ∨ 𝑧 = ∅) → (𝑦 ∩ 𝑧) = ∅)
69	62, 68	syl6 35	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 ∩ 𝑧) = ∅))
70	69	orrd 881	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑃 ∈ (𝑦 ∩ 𝑧) ∨ (𝑦 ∩ 𝑧) = ∅))
71		eleq2 2874	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑦 ∩ 𝑧)))
72		eqeq1 2810	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑥 = ∅ ↔ (𝑦 ∩ 𝑧) = ∅))
73	71, 72	orbi12d 933	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ (𝑦 ∩ 𝑧) ∨ (𝑦 ∩ 𝑧) = ∅)))
74	73	elrab 3559	. . . . . . 7 ⊢ ((𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦 ∩ 𝑧) ∨ (𝑦 ∩ 𝑧) = ∅)))
75	53, 70, 74	sylanbrc 574	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
76	75	ex 399	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) → (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
77	45, 76	syl5bi 233	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}) → (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
78	77	ralrimivv 3158	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
79		pwexg 5048	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
80	79	adantr 468	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝒫 𝐴 ∈ V)
81		rabexg 5006	. . . . 5 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ V)
82	80, 81	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ V)
83		istopg 20913	. . . 4 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ V → ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})))
84	82, 83	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})))
85	36, 78, 84	mpbir2and 695	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ Top)
86		pwidg 4366	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
87	86	adantr 468	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ 𝒫 𝐴)
88		simpr 473	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ 𝐴)
89	88	orcd 891	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑃 ∈ 𝐴 ∨ 𝐴 = ∅))
90		eleq2 2874	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝐴))
91		eqeq1 2810	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
92	90, 91	orbi12d 933	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝐴 ∨ 𝐴 = ∅)))
93	92	elrab 3559	. . . . 5 ⊢ (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝐴 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝐴 ∨ 𝐴 = ∅)))
94	87, 89, 93	sylanbrc 574	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
95		elssuni 4661	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
96	94, 95	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
97		ssrab2 3884	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴
98		sspwuni 4803	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝐴)
99	97, 98	mpbi 221	. . . 4 ⊢ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝐴
100	99	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝐴)
101	96, 100	eqssd 3815	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
102		istopon 20930	. 2 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴) ↔ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ Top ∧ 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
103	85, 101, 102	sylanbrc 574	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   ∨ wo 865  ∀wal 1635   = wceq 1637  ∃wex 1859   ∈ wcel 2156  ∀wral 3096  ∃wrex 3097  {crab 3100  Vcvv 3391   ∩ cin 3768   ⊆ wss 3769  ∅c0 4116  𝒫 cpw 4351  ∪ cuni 4630  ‘cfv 6101  Topctop 20911  TopOnctopon 20928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6064  df-fun 6103  df-fv 6109  df-top 20912  df-topon 20929

This theorem is referenced by:  pptbas  21026