Theorem clscon 21043

Description: The closure of a connected set is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
clscon	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) → (𝐽 ↾t ((cls‘𝐽)‘𝐴)) ∈ Con)

Proof of Theorem clscon

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

Step	Hyp	Ref	Expression
1		simpll3 1095	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴)))) → (𝐽 ↾t 𝐴) ∈ Con)
2		simpll1 1093	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → 𝐽 ∈ (TopOn‘𝑋))
3		simpll2 1094	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → 𝐴 ⊆ 𝑋)
4		simplrl 796	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → 𝑥 ∈ 𝐽)
5		simplrr 797	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → 𝑦 ∈ 𝐽)
6		simprl1 1099	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → (𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅)
7		n0 3890	. . . . . . . . 9 ⊢ ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴)))
8	6, 7	sylib 207	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → ∃𝑧 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴)))
9	2	adantr 480	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ (TopOn‘𝑋))
10		topontop 20541	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ Top)
12	3	adantr 480	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝐴 ⊆ 𝑋)
13		toponuni 20542	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
14	9, 13	syl 17	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝑋 = ∪ 𝐽)
15	12, 14	sseqtrd 3604	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝐴 ⊆ ∪ 𝐽)
16		inss2 3796	. . . . . . . . . 10 ⊢ (𝑥 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐽)‘𝐴)
17		simpr 476	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴)))
18	16, 17	sseldi 3566	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧 ∈ ((cls‘𝐽)‘𝐴))
19	4	adantr 480	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝑥 ∈ 𝐽)
20		inss1 3795	. . . . . . . . . 10 ⊢ (𝑥 ∩ ((cls‘𝐽)‘𝐴)) ⊆ 𝑥
21	20, 17	sseldi 3566	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧 ∈ 𝑥)
22		eqid 2610	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
23	22	clsndisj 20689	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑧 ∈ 𝑥)) → (𝑥 ∩ 𝐴) ≠ ∅)
24	11, 15, 18, 19, 21, 23	syl32anc 1326	. . . . . . . 8 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → (𝑥 ∩ 𝐴) ≠ ∅)
25	8, 24	exlimddv 1850	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → (𝑥 ∩ 𝐴) ≠ ∅)
26		simprl2 1100	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅)
27		n0 3890	. . . . . . . . 9 ⊢ ((𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴)))
28	26, 27	sylib 207	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → ∃𝑧 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴)))
29	2	adantr 480	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ (TopOn‘𝑋))
30	29, 10	syl 17	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ Top)
31	3	adantr 480	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝐴 ⊆ 𝑋)
32	29, 13	syl 17	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝑋 = ∪ 𝐽)
33	31, 32	sseqtrd 3604	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝐴 ⊆ ∪ 𝐽)
34		inss2 3796	. . . . . . . . . 10 ⊢ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐽)‘𝐴)
35		simpr 476	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴)))
36	34, 35	sseldi 3566	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧 ∈ ((cls‘𝐽)‘𝐴))
37	5	adantr 480	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝑦 ∈ 𝐽)
38		inss1 3795	. . . . . . . . . 10 ⊢ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ⊆ 𝑦
39	38, 35	sseldi 3566	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧 ∈ 𝑦)
40	22	clsndisj 20689	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑧 ∈ 𝑦)) → (𝑦 ∩ 𝐴) ≠ ∅)
41	30, 33, 36, 37, 39, 40	syl32anc 1326	. . . . . . . 8 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → (𝑦 ∩ 𝐴) ≠ ∅)
42	28, 41	exlimddv 1850	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → (𝑦 ∩ 𝐴) ≠ ∅)
43		simprl3 1101	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
44	2, 10	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → 𝐽 ∈ Top)
45	2, 13	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → 𝑋 = ∪ 𝐽)
46	3, 45	sseqtrd 3604	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → 𝐴 ⊆ ∪ 𝐽)
47	22	sscls 20670	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
48	44, 46, 47	syl2anc 691	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
49	48	sscond 3709	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → (𝑋 ∖ ((cls‘𝐽)‘𝐴)) ⊆ (𝑋 ∖ 𝐴))
50	43, 49	sstrd 3578	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ 𝐴))
51		ssv 3588	. . . . . . . . . 10 ⊢ 𝑋 ⊆ V
52		ssdif 3707	. . . . . . . . . 10 ⊢ (𝑋 ⊆ V → (𝑋 ∖ 𝐴) ⊆ (V ∖ 𝐴))
53	51, 52	ax-mp 5	. . . . . . . . 9 ⊢ (𝑋 ∖ 𝐴) ⊆ (V ∖ 𝐴)
54	50, 53	syl6ss 3580	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → (𝑥 ∩ 𝑦) ⊆ (V ∖ 𝐴))
55		disj2 3976	. . . . . . . 8 ⊢ (((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅ ↔ (𝑥 ∩ 𝑦) ⊆ (V ∖ 𝐴))
56	54, 55	sylibr 223	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → ((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅)
57		simprr 792	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))
58	48, 57	sstrd 3578	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → 𝐴 ⊆ (𝑥 ∪ 𝑦))
59	2, 3, 4, 5, 25, 42, 56, 58	nconsubb 21036	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))) → ¬ (𝐽 ↾t 𝐴) ∈ Con)
60	59	expr 641	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴)))) → (((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦) → ¬ (𝐽 ↾t 𝐴) ∈ Con))
61	1, 60	mt2d 130	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴)))) → ¬ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))
62	61	ex 449	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) → ¬ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦)))
63	62	ralrimivva 2954	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) → ¬ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦)))
64		simp1 1054	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) → 𝐽 ∈ (TopOn‘𝑋))
65	13	sseq2d 3596	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝐽))
66	65	biimpa 500	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ 𝐽)
67	22	clsss3 20673	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝐴) ⊆ ∪ 𝐽)
68	10, 67	sylan 487	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝐴) ⊆ ∪ 𝐽)
69	66, 68	syldan 486	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ ∪ 𝐽)
70	13	adantr 480	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 = ∪ 𝐽)
71	69, 70	sseqtr4d 3605	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
72	71	3adant3 1074	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
73		connsub 21034	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝐴) ⊆ 𝑋) → ((𝐽 ↾t ((cls‘𝐽)‘𝐴)) ∈ Con ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) → ¬ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))))
74	64, 72, 73	syl2anc 691	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) → ((𝐽 ↾t ((cls‘𝐽)‘𝐴)) ∈ Con ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) → ¬ ((cls‘𝐽)‘𝐴) ⊆ (𝑥 ∪ 𝑦))))
75	63, 74	mpbird 246	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Con) → (𝐽 ↾t ((cls‘𝐽)‘𝐴)) ∈ Con)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾_t crest 15904  Topctop 20517  TopOnctopon 20518  clsccl 20632  Conccon 21024

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

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-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-iin 4458  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-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-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-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-con 21025

This theorem is referenced by:  concompcld  21047