Theorem cctop 22356

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		difeq2 4076	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ ∪ 𝑦))
2	1	breq1d 5115	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ ∪ 𝑦) ≼ ω))
3		eqeq1 2740	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 = ∅ ↔ ∪ 𝑦 = ∅))
4	2, 3	orbi12d 917	. . . . . 6 ⊢ (𝑥 = ∪ 𝑦 → (((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ ∪ 𝑦) ≼ ω ∨ ∪ 𝑦 = ∅)))
5		uniss 4873	. . . . . . . 8 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
6		ssrab2 4037	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴
7		sspwuni 5060	. . . . . . . . 9 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴)
8	6, 7	mpbi 229	. . . . . . . 8 ⊢ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴
9	5, 8	sstrdi 3956	. . . . . . 7 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ⊆ 𝐴)
10		vuniex 7676	. . . . . . . 8 ⊢ ∪ 𝑦 ∈ V
11	10	elpw 4564	. . . . . . 7 ⊢ (∪ 𝑦 ∈ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
12	9, 11	sylibr 233	. . . . . 6 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ 𝒫 𝐴)
13		uni0c 4895	. . . . . . . . . . 11 ⊢ (∪ 𝑦 = ∅ ↔ ∀𝑧 ∈ 𝑦 𝑧 = ∅)
14	13	notbii 319	. . . . . . . . . 10 ⊢ (¬ ∪ 𝑦 = ∅ ↔ ¬ ∀𝑧 ∈ 𝑦 𝑧 = ∅)
15		rexnal 3103	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝑦 ¬ 𝑧 = ∅ ↔ ¬ ∀𝑧 ∈ 𝑦 𝑧 = ∅)
16	14, 15	bitr4i 277	. . . . . . . . 9 ⊢ (¬ ∪ 𝑦 = ∅ ↔ ∃𝑧 ∈ 𝑦 ¬ 𝑧 = ∅)
17		ssel2 3939	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
18		difeq2 4076	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑧 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑧))
19	18	breq1d 5115	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ 𝑧) ≼ ω))
20		eqeq1 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → (𝑥 = ∅ ↔ 𝑧 = ∅))
21	19, 20	orbi12d 917	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅)))
22	21	elrab 3645	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅)))
23	17, 22	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) → (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅)))
24	23	simprd 496	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) → ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅))
25	24	ord 862	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) → (¬ (𝐴 ∖ 𝑧) ≼ ω → 𝑧 = ∅))
26	25	con1d 145	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) → (¬ 𝑧 = ∅ → (𝐴 ∖ 𝑧) ≼ ω))
27	26	imp 407	. . . . . . . . . . 11 ⊢ (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴 ∖ 𝑧) ≼ ω)
28		ctex 8903	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ 𝑧) ≼ ω → (𝐴 ∖ 𝑧) ∈ V)
29	28	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ 𝑧) ∈ V)
30		simpllr 774	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴 ∖ 𝑧) ≼ ω) → 𝑧 ∈ 𝑦)
31		elssuni 4898	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑦 → 𝑧 ⊆ ∪ 𝑦)
32		sscon 4098	. . . . . . . . . . . . . 14 ⊢ (𝑧 ⊆ ∪ 𝑦 → (𝐴 ∖ ∪ 𝑦) ⊆ (𝐴 ∖ 𝑧))
33	30, 31, 32	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ ∪ 𝑦) ⊆ (𝐴 ∖ 𝑧))
34		ssdomg 8940	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∖ 𝑧) ∈ V → ((𝐴 ∖ ∪ 𝑦) ⊆ (𝐴 ∖ 𝑧) → (𝐴 ∖ ∪ 𝑦) ≼ (𝐴 ∖ 𝑧)))
35	29, 33, 34	sylc 65	. . . . . . . . . . . 12 ⊢ ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ ∪ 𝑦) ≼ (𝐴 ∖ 𝑧))
36		domtr 8947	. . . . . . . . . . . 12 ⊢ (((𝐴 ∖ ∪ 𝑦) ≼ (𝐴 ∖ 𝑧) ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ ∪ 𝑦) ≼ ω)
37	35, 36	sylancom 588	. . . . . . . . . . 11 ⊢ ((((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ ∪ 𝑦) ≼ ω)
38	27, 37	mpdan 685	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∧ 𝑧 ∈ 𝑦) ∧ ¬ 𝑧 = ∅) → (𝐴 ∖ ∪ 𝑦) ≼ ω)
39	38	rexlimdva2 3154	. . . . . . . . 9 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → (∃𝑧 ∈ 𝑦 ¬ 𝑧 = ∅ → (𝐴 ∖ ∪ 𝑦) ≼ ω))
40	16, 39	biimtrid 241	. . . . . . . 8 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ ∪ 𝑦 = ∅ → (𝐴 ∖ ∪ 𝑦) ≼ ω))
41	40	con1d 145	. . . . . . 7 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → (¬ (𝐴 ∖ ∪ 𝑦) ≼ ω → ∪ 𝑦 = ∅))
42	41	orrd 861	. . . . . 6 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ((𝐴 ∖ ∪ 𝑦) ≼ ω ∨ ∪ 𝑦 = ∅))
43	4, 12, 42	elrabd 3647	. . . . 5 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
44	43	ax-gen 1797	. . . 4 ⊢ ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
45		difeq2 4076	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑦))
46	45	breq1d 5115	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ 𝑦) ≼ ω))
47		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
48	46, 47	orbi12d 917	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ 𝑦) ≼ ω ∨ 𝑦 = ∅)))
49	48	elrab 3645	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑦) ≼ ω ∨ 𝑦 = ∅)))
50		ssinss1 4197	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝐴 → (𝑦 ∩ 𝑧) ⊆ 𝐴)
51		vex 3449	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
52	51	elpw 4564	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
53	51	inex1 5274	. . . . . . . . . . 11 ⊢ (𝑦 ∩ 𝑧) ∈ V
54	53	elpw 4564	. . . . . . . . . 10 ⊢ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ↔ (𝑦 ∩ 𝑧) ⊆ 𝐴)
55	50, 52, 54	3imtr4i 291	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝐴 → (𝑦 ∩ 𝑧) ∈ 𝒫 𝐴)
56	55	ad2antrr 724	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅))) → (𝑦 ∩ 𝑧) ∈ 𝒫 𝐴)
57		difindi 4241	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ (𝑦 ∩ 𝑧)) = ((𝐴 ∖ 𝑦) ∪ (𝐴 ∖ 𝑧))
58		unctb 10141	. . . . . . . . . . . 12 ⊢ (((𝐴 ∖ 𝑦) ≼ ω ∧ (𝐴 ∖ 𝑧) ≼ ω) → ((𝐴 ∖ 𝑦) ∪ (𝐴 ∖ 𝑧)) ≼ ω)
59	57, 58	eqbrtrid 5140	. . . . . . . . . . 11 ⊢ (((𝐴 ∖ 𝑦) ≼ ω ∧ (𝐴 ∖ 𝑧) ≼ ω) → (𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω)
60	59	orcd 871	. . . . . . . . . 10 ⊢ (((𝐴 ∖ 𝑦) ≼ ω ∧ (𝐴 ∖ 𝑧) ≼ ω) → ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅))
61		ineq1 4165	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝑧) = (∅ ∩ 𝑧))
62		0in 4353	. . . . . . . . . . . 12 ⊢ (∅ ∩ 𝑧) = ∅
63	61, 62	eqtrdi 2792	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∩ 𝑧) = ∅)
64	63	olcd 872	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅))
65		ineq2 4166	. . . . . . . . . . . 12 ⊢ (𝑧 = ∅ → (𝑦 ∩ 𝑧) = (𝑦 ∩ ∅))
66		in0 4351	. . . . . . . . . . . 12 ⊢ (𝑦 ∩ ∅) = ∅
67	65, 66	eqtrdi 2792	. . . . . . . . . . 11 ⊢ (𝑧 = ∅ → (𝑦 ∩ 𝑧) = ∅)
68	67	olcd 872	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅))
69	60, 64, 68	ccase2 1038	. . . . . . . . 9 ⊢ ((((𝐴 ∖ 𝑦) ≼ ω ∨ 𝑦 = ∅) ∧ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅)) → ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅))
70	69	ad2ant2l 744	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑦) ≼ ω ∨ 𝑦 = ∅)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ 𝑧) ≼ ω ∨ 𝑧 = ∅))) → ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅))
71	56, 70	jca 512	. . . . . . 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) = ∅)))
72	49, 22, 71	syl2anb 598	. . . . . 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) = ∅)))
73		difeq2 4076	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝑦 ∩ 𝑧)))
74	73	breq1d 5115	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω))
75		eqeq1 2740	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑥 = ∅ ↔ (𝑦 ∩ 𝑧) = ∅))
76	74, 75	orbi12d 917	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅)))
77	76	elrab 3645	. . . . . 6 ⊢ ((𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ↔ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ ((𝐴 ∖ (𝑦 ∩ 𝑧)) ≼ ω ∨ (𝑦 ∩ 𝑧) = ∅)))
78	72, 77	sylibr 233	. . . . 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 = ∅)})
79	78	rgen2 3194	. . . 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 = ∅)}
80	44, 79	pm3.2i 471	. . 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 = ∅)})
81		pwexg 5333	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
82		rabexg 5288	. . . 4 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ V)
83		istopg 22244	. . . 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 = ∅)})))
84	81, 82, 83	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 = ∅)})))
85	80, 84	mpbiri 257	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ Top)
86		difeq2 4076	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐴))
87		difid 4330	. . . . . . . 8 ⊢ (𝐴 ∖ 𝐴) = ∅
88	86, 87	eqtrdi 2792	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 ∖ 𝑥) = ∅)
89	88	breq1d 5115	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐴 ∖ 𝑥) ≼ ω ↔ ∅ ≼ ω))
90		eqeq1 2740	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = ∅ ↔ 𝐴 = ∅))
91	89, 90	orbi12d 917	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅) ↔ (∅ ≼ ω ∨ 𝐴 = ∅)))
92		pwidg 4580	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
93		omex 9579	. . . . . . . 8 ⊢ ω ∈ V
94	93	0dom 9050	. . . . . . 7 ⊢ ∅ ≼ ω
95	94	orci 863	. . . . . 6 ⊢ (∅ ≼ ω ∨ 𝐴 = ∅)
96	95	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∅ ≼ ω ∨ 𝐴 = ∅))
97	91, 92, 96	elrabd 3647	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
98		elssuni 4898	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
99	97, 98	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
100	8	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ⊆ 𝐴)
101	99, 100	eqssd 3961	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)})
102		istopon 22261	. 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 = ∅)}))
103	85, 101, 102	sylanbrc 583	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝒫 𝐴 ∣ ((𝐴 ∖ 𝑥) ≼ ω ∨ 𝑥 = ∅)} ∈ (TopOn‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ∖ cdif 3907   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  ∪ cuni 4865   class class class wbr 5105  ‘cfv 6496  ωcom 7802   ≼ cdom 8881  Topctop 22242  TopOnctopon 22259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9446  df-dju 9837  df-card 9875  df-top 22243  df-topon 22260

This theorem is referenced by: (None)