ppttop - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ppttop		Structured version Visualization version GIF version

Theorem ppttop 22510

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 4071	. . . . 5 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)))
2		eleq2 2823	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ ∪ 𝑦))
3		eqeq1 2737	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 = ∅ ↔ ∪ 𝑦 = ∅))
4	2, 3	orbi12d 918	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ ∪ 𝑦 ∨ ∪ 𝑦 = ∅)))
5		simprl 770	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → 𝑦 ⊆ 𝒫 𝐴)
6		sspwuni 5104	. . . . . . . . 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 7729	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
9	8	elpw 4607	. . . . . . . 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 4346	. . . . . . . . . 10 ⊢ (¬ ∪ 𝑦 = ∅ ↔ ∃𝑧 𝑧 ∈ ∪ 𝑦)
12		eluni2 4913	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥)
13		r19.29 3115	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥) → ∃𝑥 ∈ 𝑦 ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥))
14		n0i 4334	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑥 → ¬ 𝑥 = ∅)
15	14	adantl 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → ¬ 𝑥 = ∅)
16		simpl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))
17	16	ord 863	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → (¬ 𝑃 ∈ 𝑥 → 𝑥 = ∅))
18	15, 17	mt3d 148	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥) → 𝑃 ∈ 𝑥)
19		simpl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥)) → 𝑥 ∈ 𝑦)
20		elunii 4914	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) → 𝑃 ∈ ∪ 𝑦)
21	18, 19, 20	syl2an2 685	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ∧ 𝑧 ∈ 𝑥)) → 𝑃 ∈ ∪ 𝑦)
22	21	rexlimiva 3148	. . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) → (∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥 → 𝑃 ∈ ∪ 𝑦))
25	24	ad2antll 728	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (∃𝑥 ∈ 𝑦 𝑧 ∈ 𝑥 → 𝑃 ∈ ∪ 𝑦))
26	12, 25	biimtrid 241	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (𝑧 ∈ ∪ 𝑦 → 𝑃 ∈ ∪ 𝑦))
27	26	exlimdv 1937	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (∃𝑧 𝑧 ∈ ∪ 𝑦 → 𝑃 ∈ ∪ 𝑦))
28	11, 27	biimtrid 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 862	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → (𝑃 ∈ ∪ 𝑦 ∨ ∪ 𝑦 = ∅))
31	4, 10, 30	elrabd 3686	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅))) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
32	31	ex 414	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ((𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
33	1, 32	biimtrid 241	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
34	33	alrimiv 1931	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}))
35		eleq2 2823	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
36		eqeq1 2737	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
37	35, 36	orbi12d 918	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)))
38	37	elrab 3684	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)))
39		eleq2 2823	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧))
40		eqeq1 2737	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
41	39, 40	orbi12d 918	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
42	41	elrab 3684	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
43	38, 42	anbi12i 628	. . . . 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 2823	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑦 ∩ 𝑧)))
45		eqeq1 2737	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑥 = ∅ ↔ (𝑦 ∩ 𝑧) = ∅))
46	44, 45	orbi12d 918	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ (𝑦 ∩ 𝑧) ∨ (𝑦 ∩ 𝑧) = ∅)))
47		inss1 4229	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ⊆ 𝑦
48		simprll 778	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → 𝑦 ∈ 𝒫 𝐴)
49	48	elpwid 4612	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → 𝑦 ⊆ 𝐴)
50	47, 49	sstrid 3994	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑦 ∩ 𝑧) ⊆ 𝐴)
51		vex 3479	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
52	51	inex1 5318	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ∈ V
53	52	elpw 4607	. . . . . . . 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 981	. . . . . . . . . . 11 ⊢ (¬ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧) ↔ (¬ 𝑃 ∈ 𝑦 ∨ ¬ 𝑃 ∈ 𝑧))
56		elin 3965	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (𝑦 ∩ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧))
57	55, 56	xchnxbir 333	. . . . . . . . . 10 ⊢ (¬ 𝑃 ∈ (𝑦 ∩ 𝑧) ↔ (¬ 𝑃 ∈ 𝑦 ∨ ¬ 𝑃 ∈ 𝑧))
58		simprlr 779	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅))
59	58	ord 863	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ 𝑦 → 𝑦 = ∅))
60		simprrr 781	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))
61	60	ord 863	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))) → (¬ 𝑃 ∈ 𝑧 → 𝑧 = ∅))
62	59, 61	orim12d 964	. . . . . . . . . 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	biimtrid 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 4239	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∨ 𝑧 ⊆ ∅) → (𝑦 ∩ 𝑧) ⊆ ∅)
65		ss0b 4398	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ∅ ↔ 𝑦 = ∅)
66		ss0b 4398	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ ∅ ↔ 𝑧 = ∅)
67	65, 66	orbi12i 914	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ ∅ ∨ 𝑧 ⊆ ∅) ↔ (𝑦 = ∅ ∨ 𝑧 = ∅))
68		ss0b 4398	. . . . . . . . . 10 ⊢ ((𝑦 ∩ 𝑧) ⊆ ∅ ↔ (𝑦 ∩ 𝑧) = ∅)
69	64, 67, 68	3imtr3i 291	. . . . . . . . 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 862	. . . . . . 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 3686	. . . . . 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 414	. . . . 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	biimtrid 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 3199	. . 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 5377	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
77	76	adantr 482	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝒫 𝐴 ∈ V)
78		rabexg 5332	. . . 4 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ V)
79		istopg 22397	. . . 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 712	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ Top)
82		eleq2 2823	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝐴))
83		eqeq1 2737	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
84	82, 83	orbi12d 918	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑃 ∈ 𝑥 ∨ 𝑥 = ∅) ↔ (𝑃 ∈ 𝐴 ∨ 𝐴 = ∅)))
85		pwidg 4623	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
86	85	adantr 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ 𝒫 𝐴)
87		animorrl 980	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑃 ∈ 𝐴 ∨ 𝐴 = ∅))
88	84, 86, 87	elrabd 3686	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
89		elssuni 4942	. . . 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 4078	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴
92		sspwuni 5104	. . . . 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 4000	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)})
96		istopon 22414	. 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 584	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 ∪ cuni 4909 ‘cfv 6544 Topctop 22395 TopOnctopon 22412

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-top 22396 df-topon 22413

This theorem is referenced by: pptbas 22511

W3C validator