Theorem ppttop 22185

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 4009	. . . . 5 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)))
2		eleq2 2822	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ ∪ 𝑦))
3		eqeq1 2737	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 = ∅ ↔ ∪ 𝑦 = ∅))
4	2, 3	orbi12d 915	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ ∪ 𝑦 ∨ ∪ 𝑦 = ∅)))
5		simprl 767	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → 𝑦 ⊆ 𝒫 𝐴)
6		sspwuni 5032	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
7	5, 6	sylib 217	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → ∪ 𝑦 ⊆ 𝐴)
8		vuniex 7612	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
9	8	elpw 4540	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
10	7, 9	sylibr 233	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → ∪ 𝑦 ∈ 𝒫 𝐴)
11		neq0 4282	. . . . . . . . . 10 ⊢ (¬ ∪ 𝑦 = ∅ ↔ ∃𝑧 𝑧 ∈ ∪ 𝑦)
12		eluni2 4845	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥)
13		r19.29 3111	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥) → ∃𝑥 ∈ 𝑦 ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥))
14		n0i 4270	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑥 → ¬ 𝑥 = ∅)
15	14	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → ¬ 𝑥 = ∅)
16		simpl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))
17	16	ord 860	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → (¬ 𝑃 ∈ 𝑥 → 𝑥 = ∅))
18	15, 17	mt3d 148	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → 𝑃 ∈ 𝑥)
19		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥)) → 𝑥 ∈ 𝑦)
20		elunii 4846	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) → 𝑃 ∈ ∪ 𝑦)
21	18, 19, 20	syl2an2 682	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥)) → 𝑃 ∈ ∪ 𝑦)
22	21	rexlimiva 3138	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ 𝑦 ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → 𝑃 ∈ ∪ 𝑦)
23	13, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥) → 𝑃 ∈ ∪ 𝑦)
24	23	ex 412	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) → (∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥 → 𝑃 ∈ ∪ 𝑦))
25	24	ad2antll 725	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥 → 𝑃 ∈ ∪ 𝑦))
26	12, 25	syl5bi 241	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (𝑧 ∈ ∪ 𝑦 → 𝑃 ∈ ∪ 𝑦))
27	26	exlimdv 1932	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (∃𝑧 𝑧 ∈ ∪ 𝑦 → 𝑃 ∈ ∪ 𝑦))
28	11, 27	syl5bi 241	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (¬ ∪ 𝑦 = ∅ → 𝑃 ∈ ∪ 𝑦))
29	28	con1d 145	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (¬ 𝑃 ∈ ∪ 𝑦 → ∪ 𝑦 = ∅))
30	29	orrd 859	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (𝑃 ∈ ∪ 𝑦 ∨ ∪ 𝑦 = ∅))
31	4, 10, 30	elrabd 3628	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
32	31	ex 412	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ((𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
33	1, 32	syl5bi 241	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
34	33	alrimiv 1926	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
35		eleq2 2822	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
36		eqeq1 2737	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
37	35, 36	orbi12d 915	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)))
38	37	elrab 3626	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)))
39		eleq2 2822	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧))
40		eqeq1 2737	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
41	39, 40	orbi12d 915	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
42	41	elrab 3626	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
43	38, 42	anbi12i 626	. . . . 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 = ∅))))
44		eleq2 2822	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑦 ∩ 𝑧)))
45		eqeq1 2737	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑥 = ∅ ↔ (𝑦 ∩ 𝑧) = ∅))
46	44, 45	orbi12d 915	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ (𝑦 ∩ 𝑧) ∨ (𝑦 ∩ 𝑧) = ∅)))
47		inss1 4165	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ⊆ 𝑦
48		simprll 775	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → 𝑦 ∈ 𝒫 𝐴)
49	48	elpwid 4547	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → 𝑦 ⊆ 𝐴)
50	47, 49	sstrid 3934	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑦 ∩ 𝑧) ⊆ 𝐴)
51		vex 3438	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
52	51	inex1 5244	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ∈ V
53	52	elpw 4540	. . . . . . . 8 ⊢ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ↔ (𝑦 ∩ 𝑧) ⊆ 𝐴)
54	50, 53	sylibr 233	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑦 ∩ 𝑧) ∈ 𝒫 𝐴)
55		ianor 978	. . . . . . . . . . 11 ⊢ (¬ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧) ↔ (¬ 𝑃 ∈ 𝑦 ∨ ¬ 𝑃 ∈ 𝑧))
56		elin 3905	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (𝑦 ∩ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧))
57	55, 56	xchnxbir 332	. . . . . . . . . 10 ⊢ (¬ 𝑃 ∈ (𝑦 ∩ 𝑧) ↔ (¬ 𝑃 ∈ 𝑦 ∨ ¬ 𝑃 ∈ 𝑧))
58		simprlr 776	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅))
59	58	ord 860	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ 𝑦 → 𝑦 = ∅))
60		simprrr 778	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))
61	60	ord 860	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ 𝑧 → 𝑧 = ∅))
62	59, 61	orim12d 961	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → ((¬ 𝑃 ∈ 𝑦 ∨ ¬ 𝑃 ∈ 𝑧) → (𝑦 = ∅ ∨ 𝑧 = ∅)))
63	57, 62	syl5bi 241	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 = ∅ ∨ 𝑧 = ∅)))
64		inss 4175	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∨ 𝑧 ⊆ ∅) → (𝑦 ∩ 𝑧) ⊆ ∅)
65		ss0b 4334	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
66		ss0b 4334	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ ∅ ↔ 𝑧 = ∅)
67	65, 66	orbi12i 911	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∨ 𝑧 ⊆ ∅) ↔ (𝑦 = ∅ ∨ 𝑧 = ∅))
68		ss0b 4334	. . . . . . . . . 10 ⊢ ((𝑦 ∩ 𝑧) ⊆ ∅ ↔ (𝑦 ∩ 𝑧) = ∅)
69	64, 67, 68	3imtr3i 290	. . . . . . . . 9 ⊢ ((𝑦 = ∅ ∨ 𝑧 = ∅) → (𝑦 ∩ 𝑧) = ∅)
70	63, 69	syl6 35	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 ∩ 𝑧) = ∅))
71	70	orrd 859	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑃 ∈ (𝑦 ∩ 𝑧) ∨ (𝑦 ∩ 𝑧) = ∅))
72	46, 54, 71	elrabd 3628	. . . . . 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 = ∅)})
73	72	ex 412	. . . . 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 = ∅)}))
74	43, 73	syl5bi 241	. . . 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 = ∅)}))
75	74	ralrimivv 3189	. . 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 = ∅)})
76		pwexg 5304	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
77	76	adantr 480	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝒫 𝐴 ∈ V)
78		rabexg 5258	. . . 4 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ V)
79		istopg 22072	. . . 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 = ∅)})))
80	77, 78, 79	3syl 18	. . 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 = ∅)})))
81	34, 75, 80	mpbir2and 709	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ Top)
82		eleq2 2822	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝐴))
83		eqeq1 2737	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
84	82, 83	orbi12d 915	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝐴 ∨ 𝐴 = ∅)))
85		pwidg 4558	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
86	85	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ 𝒫 𝐴)
87		animorrl 977	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑃 ∈ 𝐴 ∨ 𝐴 = ∅))
88	84, 86, 87	elrabd 3628	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
89		elssuni 4874	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
90	88, 89	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
91		ssrab2 4016	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴
92		sspwuni 5032	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝐴)
93	91, 92	mpbi 229	. . . 4 ⊢ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝐴
94	93	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝐴)
95	90, 94	eqssd 3940	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
96		istopon 22089	. 2 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴) ↔ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ Top ∧ 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
97	81, 95, 96	sylanbrc 582	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2101  ∀wral 3059  ∃wrex 3068  {crab 3221  Vcvv 3434   ∩ cin 3888   ⊆ wss 3889  ∅c0 4259  𝒫 cpw 4536  ∪ cuni 4841  ‘cfv 6447  Topctop 22070  TopOnctopon 22087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-iota 6399  df-fun 6449  df-fv 6455  df-top 22071  df-topon 22088

This theorem is referenced by:  pptbas  22186