Theorem ppttop 21299

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 3972	. . . . 5 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)))
2		eleq2 2870	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ ∪ 𝑦))
3		eqeq1 2798	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 = ∅ ↔ ∪ 𝑦 = ∅))
4	2, 3	orbi12d 913	. . . . . . 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 4923	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
7	5, 6	sylib 219	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → ∪ 𝑦 ⊆ 𝐴)
8		vuniex 7327	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
9	8	elpw 4461	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
10	7, 9	sylibr 235	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → ∪ 𝑦 ∈ 𝒫 𝐴)
11		neq0 4231	. . . . . . . . . 10 ⊢ (¬ ∪ 𝑦 = ∅ ↔ ∃𝑧 𝑧 ∈ ∪ 𝑦)
12		eluni2 4751	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥)
13		r19.29 3217	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥) → ∃𝑥 ∈ 𝑦 ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥))
14		n0i 4221	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑥 → ¬ 𝑥 = ∅)
15	14	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → ¬ 𝑥 = ∅)
16		simpl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))
17	16	ord 859	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → (¬ 𝑃 ∈ 𝑥 → 𝑥 = ∅))
18	15, 17	mt3d 150	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → 𝑃 ∈ 𝑥)
19		simpl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥)) → 𝑥 ∈ 𝑦)
20		elunii 4752	. . . . . . . . . . . . . . . . 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 3243	. . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . 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 243	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (𝑧 ∈ ∪ 𝑦 → 𝑃 ∈ ∪ 𝑦))
27	26	exlimdv 1912	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (∃𝑧 𝑧 ∈ ∪ 𝑦 → 𝑃 ∈ ∪ 𝑦))
28	11, 27	syl5bi 243	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (¬ ∪ 𝑦 = ∅ → 𝑃 ∈ ∪ 𝑦))
29	28	con1d 147	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (¬ 𝑃 ∈ ∪ 𝑦 → ∪ 𝑦 = ∅))
30	29	orrd 858	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (𝑃 ∈ ∪ 𝑦 ∨ ∪ 𝑦 = ∅))
31	4, 10, 30	elrabd 3619	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
32	31	ex 413	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ((𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
33	1, 32	syl5bi 243	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
34	33	alrimiv 1906	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
35		eleq2 2870	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
36		eqeq1 2798	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
37	35, 36	orbi12d 913	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)))
38	37	elrab 3617	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)))
39		eleq2 2870	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧))
40		eqeq1 2798	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
41	39, 40	orbi12d 913	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
42	41	elrab 3617	. . . . . 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 2870	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑦 ∩ 𝑧)))
45		eqeq1 2798	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑥 = ∅ ↔ (𝑦 ∩ 𝑧) = ∅))
46	44, 45	orbi12d 913	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ (𝑦 ∩ 𝑧) ∨ (𝑦 ∩ 𝑧) = ∅)))
47		inss1 4127	. . . . . . . . 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 4467	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → 𝑦 ⊆ 𝐴)
50	47, 49	sstrid 3902	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑦 ∩ 𝑧) ⊆ 𝐴)
51		vex 3439	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
52	51	inex1 5115	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ∈ V
53	52	elpw 4461	. . . . . . . 8 ⊢ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ↔ (𝑦 ∩ 𝑧) ⊆ 𝐴)
54	50, 53	sylibr 235	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑦 ∩ 𝑧) ∈ 𝒫 𝐴)
55		ianor 976	. . . . . . . . . . 11 ⊢ (¬ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧) ↔ (¬ 𝑃 ∈ 𝑦 ∨ ¬ 𝑃 ∈ 𝑧))
56		elin 4092	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (𝑦 ∩ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧))
57	55, 56	xchnxbir 334	. . . . . . . . . 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 859	. . . . . . . . . . 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 859	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ 𝑧 → 𝑧 = ∅))
62	59, 61	orim12d 959	. . . . . . . . . 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 243	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 = ∅ ∨ 𝑧 = ∅)))
64		inss 4137	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∨ 𝑧 ⊆ ∅) → (𝑦 ∩ 𝑧) ⊆ ∅)
65		ss0b 4273	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
66		ss0b 4273	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ ∅ ↔ 𝑧 = ∅)
67	65, 66	orbi12i 909	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∨ 𝑧 ⊆ ∅) ↔ (𝑦 = ∅ ∨ 𝑧 = ∅))
68		ss0b 4273	. . . . . . . . . 10 ⊢ ((𝑦 ∩ 𝑧) ⊆ ∅ ↔ (𝑦 ∩ 𝑧) = ∅)
69	64, 67, 68	3imtr3i 292	. . . . . . . . 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 858	. . . . . . 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 3619	. . . . . 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 413	. . . . 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 243	. . . 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 3156	. . 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 5173	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
77	76	adantr 481	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝒫 𝐴 ∈ V)
78		rabexg 5128	. . . 4 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ V)
79		istopg 21187	. . . 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 2870	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝐴))
83		eqeq1 2798	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
84	82, 83	orbi12d 913	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝐴 ∨ 𝐴 = ∅)))
85		pwidg 4470	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
86	85	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ 𝒫 𝐴)
87		animorrl 975	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑃 ∈ 𝐴 ∨ 𝐴 = ∅))
88	84, 86, 87	elrabd 3619	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
89		elssuni 4776	. . . 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 3979	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴
92		sspwuni 4923	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝐴)
93	91, 92	mpbi 231	. . . 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 3908	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
96		istopon 21204	. 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 583	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842  ∀wal 1520   = wceq 1522  ∃wex 1762   ∈ wcel 2080  ∀wral 3104  ∃wrex 3105  {crab 3108  Vcvv 3436   ∩ cin 3860   ⊆ wss 3861  ∅c0 4213  𝒫 cpw 4455  ∪ cuni 4747  ‘cfv 6228  Topctop 21185  TopOnctopon 21202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-iota 6192  df-fun 6230  df-fv 6236  df-top 21186  df-topon 21203

This theorem is referenced by:  pptbas  21300