Theorem cctop 20620

Description: The countable complement topology on a set 𝐴. Example 4 in [Munkres] p. 77. (Contributed by FL, 23-Aug-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
cctop	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem cctop

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniss 4394	. . . . . . . 8 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
2		ssrab2 3650	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴
3		sspwuni 4547	. . . . . . . . 9 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴)
4	2, 3	mpbi 219	. . . . . . . 8 ⊢ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴
5	1, 4	syl6ss 3580	. . . . . . 7 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ⊆ 𝐴)
6		vuniex 6852	. . . . . . . 8 ⊢ ∪ 𝑦 ∈ V
7	6	elpw 4114	. . . . . . 7 ⊢ (∪ 𝑦 ∈ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
8	5, 7	sylibr 223	. . . . . 6 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ 𝒫 𝐴)
9		uni0c 4400	. . . . . . . . . . 11 ⊢ (∪ 𝑦 = ∅ ↔ ∀𝑧 ∈ 𝑦 𝑧 = ∅)
10	9	notbii 309	. . . . . . . . . 10 ⊢ (¬ ∪ 𝑦 = ∅ ↔ ¬ ∀𝑧 ∈ 𝑦 𝑧 = ∅)
11		rexnal 2978	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑦 ¬ 𝑧 = ∅ ↔ ¬ ∀𝑧 ∈ 𝑦 𝑧 = ∅)
12	10, 11	bitr4i 266	. . . . . . . . 9 ⊢ (¬ ∪ 𝑦 = ∅ ↔ ∃𝑧 ∈ 𝑦 ¬ 𝑧 = ∅)
13		ssel2 3563	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
14		difeq2 3684	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑧))
15	14	breq1d 4593	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ 𝑧) ≼ ω))
16		eqeq1 2614	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
17	15, 16	orbi12d 742	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → (((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅)))
18	17	elrab 3331	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅)))
19	13, 18	sylib 207	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅)))
20	19	simprd 478	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) → ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅))
21	20	ord 391	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) → (¬ (𝐴 ∖ 𝑧) ≼ ω → 𝑧 = ∅))
22	21	con1d 138	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) → (¬ 𝑧 = ∅ → (𝐴 ∖ 𝑧) ≼ ω))
23	22	imp 444	. . . . . . . . . . . 12 ⊢ (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴 ∖ 𝑧) ≼ ω)
24		ctex 7856	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∖ 𝑧) ≼ ω → (𝐴 ∖ 𝑧) ∈ V)
25	24	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ 𝑧) ∈ V)
26		simpllr 795	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴 ∖ 𝑧) ≼ ω) → 𝑧 ∈ 𝑦)
27		elssuni 4403	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑦 → 𝑧 ⊆ ∪ 𝑦)
28		sscon 3706	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ ∪ 𝑦 → (𝐴 ∖ ∪ 𝑦) ⊆ (𝐴 ∖ 𝑧))
29	26, 27, 28	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ ∪ 𝑦) ⊆ (𝐴 ∖ 𝑧))
30		ssdomg 7887	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ 𝑧) ∈ V → ((𝐴 ∖ ∪ 𝑦) ⊆ (𝐴 ∖ 𝑧) → (𝐴 ∖ ∪ 𝑦) ≼ (𝐴 ∖ 𝑧)))
31	25, 29, 30	sylc 63	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ ∪ 𝑦) ≼ (𝐴 ∖ 𝑧))
32		domtr 7895	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∖ ∪ 𝑦) ≼ (𝐴 ∖ 𝑧) ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ ∪ 𝑦) ≼ ω)
33	31, 32	sylancom 698	. . . . . . . . . . . 12 ⊢ ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ ∪ 𝑦) ≼ ω)
34	23, 33	mpdan 699	. . . . . . . . . . 11 ⊢ (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴 ∖ ∪ 𝑦) ≼ ω)
35	34	exp31 628	. . . . . . . . . 10 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → (𝑧 ∈ 𝑦 → (¬ 𝑧 = ∅ → (𝐴 ∖ ∪ 𝑦) ≼ ω)))
36	35	rexlimdv 3012	. . . . . . . . 9 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → (∃𝑧 ∈ 𝑦 ¬ 𝑧 = ∅ → (𝐴 ∖ ∪ 𝑦) ≼ ω))
37	12, 36	syl5bi 231	. . . . . . . 8 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ ∪ 𝑦 = ∅ → (𝐴 ∖ ∪ 𝑦) ≼ ω))
38	37	con1d 138	. . . . . . 7 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ (𝐴 ∖ ∪ 𝑦) ≼ ω → ∪ 𝑦 = ∅))
39	38	orrd 392	. . . . . 6 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ((𝐴 ∖ ∪ 𝑦) ≼ ω ∨ ∪ 𝑦 = ∅))
40		difeq2 3684	. . . . . . . . 9 ⊢ (𝑥 = ∪ 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ ∪ 𝑦))
41	40	breq1d 4593	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ ∪ 𝑦) ≼ ω))
42		eqeq1 2614	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 = ∅ ↔ ∪ 𝑦 = ∅))
43	41, 42	orbi12d 742	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → (((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ ∪ 𝑦) ≼ ω ∨ ∪ 𝑦 = ∅)))
44	43	elrab 3331	. . . . . 6 ⊢ (∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (∪ 𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ ∪ 𝑦) ≼ ω ∨ ∪ 𝑦 = ∅)))
45	8, 39, 44	sylanbrc 695	. . . . 5 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
46	45	ax-gen 1713	. . . 4 ⊢ ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
47		difeq2 3684	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
48	47	breq1d 4593	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ 𝑦) ≼ ω))
49		eqeq1 2614	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
50	48, 49	orbi12d 742	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ 𝑦) ≼ ω ∨ 𝑦 = ∅)))
51	50	elrab 3331	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑦) ≼ ω ∨ 𝑦 = ∅)))
52		ssinss1 3803	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝐴 → (𝑦 ∩ 𝑧) ⊆ 𝐴)
53		vex 3176	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
54	53	elpw 4114	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
55	53	inex1 4727	. . . . . . . . . . 11 ⊢ (𝑦 ∩ 𝑧) ∈ V
56	55	elpw 4114	. . . . . . . . . 10 ⊢ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ↔ (𝑦 ∩ 𝑧) ⊆ 𝐴)
57	52, 54, 56	3imtr4i 280	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝐴 → (𝑦 ∩ 𝑧) ∈ 𝒫 𝐴)
58	57	ad2antrr 758	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅))) → (𝑦 ∩ 𝑧) ∈ 𝒫 𝐴)
59		difindi 3840	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ (𝑦 ∩ 𝑧)) = ((𝐴 ∖ 𝑦) ∪ (𝐴 ∖ 𝑧))
60		unctb 8910	. . . . . . . . . . . 12 ⊢ (((𝐴 ∖ 𝑦) ≼ ω ∧ (𝐴 ∖ 𝑧) ≼ ω) → ((𝐴 ∖ 𝑦) ∪ (𝐴 ∖ 𝑧)) ≼ ω)
61	59, 60	syl5eqbr 4618	. . . . . . . . . . 11 ⊢ (((𝐴 ∖ 𝑦) ≼ ω ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω)
62	61	orcd 406	. . . . . . . . . 10 ⊢ (((𝐴 ∖ 𝑦) ≼ ω ∧ (𝐴 ∖ 𝑧) ≼ ω) → ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅))
63		ineq1 3769	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝑧) = (∅ ∩ 𝑧))
64		0in 3921	. . . . . . . . . . . 12 ⊢ (∅ ∩ 𝑧) = ∅
65	63, 64	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝑧) = ∅)
66	65	olcd 407	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅))
67		ineq2 3770	. . . . . . . . . . . 12 ⊢ (𝑧 = ∅ → (𝑦 ∩ 𝑧) = (𝑦 ∩ ∅))
68		in0 3920	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ ∅) = ∅
69	67, 68	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑦 ∩ 𝑧) = ∅)
70	69	olcd 407	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅))
71	62, 66, 70	ccase2 986	. . . . . . . . 9 ⊢ ((((𝐴 ∖ 𝑦) ≼ ω ∨ 𝑦 = ∅) ∧ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅)) → ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅))
72	71	ad2ant2l 778	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅))
73	58, 72	jca 553	. . . . . . 7 ⊢ (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅)))
74	51, 18, 73	syl2anb 495	. . . . . 6 ⊢ ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)}) → ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅)))
75		difeq2 3684	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝑦 ∩ 𝑧)))
76	75	breq1d 4593	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω))
77		eqeq1 2614	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑥 = ∅ ↔ (𝑦 ∩ 𝑧) = ∅))
78	76, 77	orbi12d 742	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅)))
79	78	elrab 3331	. . . . . 6 ⊢ ((𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅)))
80	74, 79	sylibr 223	. . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)}) → (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
81	80	rgen2a 2960	. . . 4 ⊢ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)}
82	46, 81	pm3.2i 470	. . 3 ⊢ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
83		pwexg 4776	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
84		rabexg 4739	. . . 4 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ V)
85		istopg 20525	. . . 4 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ V → ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})))
86	83, 84, 85	3syl 18	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})))
87	82, 86	mpbiri 247	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top)
88		pwidg 4121	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
89		omex 8423	. . . . . . . 8 ⊢ ω ∈ V
90	89	0dom 7975	. . . . . . 7 ⊢ ∅ ≼ ω
91	90	orci 404	. . . . . 6 ⊢ (∅ ≼ ω ∨ 𝐴 = ∅)
92	91	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∅ ≼ ω ∨ 𝐴 = ∅))
93		difeq2 3684	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐴))
94		difid 3902	. . . . . . . . 9 ⊢ (𝐴 ∖ 𝐴) = ∅
95	93, 94	syl6eq 2660	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐴 ∖ 𝑥) = ∅)
96	95	breq1d 4593	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐴 ∖ 𝑥) ≼ ω ↔ ∅ ≼ ω))
97		eqeq1 2614	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
98	96, 97	orbi12d 742	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ (∅ ≼ ω ∨ 𝐴 = ∅)))
99	98	elrab 3331	. . . . 5 ⊢ (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝐴 ∈ 𝒫 𝐴 ∧ (∅ ≼ ω ∨ 𝐴 = ∅)))
100	88, 92, 99	sylanbrc 695	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
101		elssuni 4403	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
102	100, 101	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
103	4	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴)
104	102, 103	eqssd 3585	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
105		istopon 20540	. 2 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴) ↔ ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top ∧ 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)}))
106	87, 104, 105	sylanbrc 695	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  ‘cfv 5804  ωcom 6957   ≼ cdom 7839  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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-cda 8873  df-top 20521  df-topon 20523

This theorem is referenced by: (None)