Theorem clsconn 21233

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

Assertion

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

Proof of Theorem clsconn

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

Step	Hyp	Ref	Expression
1		simpll3 1102	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) ∧ ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴)))) → (𝐽 ↾t 𝐴) ∈ Conn)
2		simpll1 1100	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 1101	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 800	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 801	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 1106	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3931	. . . . . . . . 9 ⊢ ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴)))
8	6, 7	sylib 208	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 481	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 20718	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 481	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 20719	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
14	9, 13	syl 17	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3641	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3834	. . . . . . . . . 10 ⊢ (𝑥 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐽)‘𝐴)
17		simpr 477	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3601	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 481	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3833	. . . . . . . . . 10 ⊢ (𝑥 ∩ ((cls‘𝐽)‘𝐴)) ⊆ 𝑥
21	20, 17	sseldi 3601	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 2622	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
23	22	clsndisj 20879	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑧 ∈ 𝑥)) → (𝑥 ∩ 𝐴) ≠ ∅)
24	11, 15, 18, 19, 21, 23	syl32anc 1334	. . . . . . . 8 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 1863	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 1107	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3931	. . . . . . . . 9 ⊢ ((𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴)))
28	26, 27	sylib 208	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 481	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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) ∈ Conn) ∧ (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 481	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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) ∈ Conn) ∧ (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 3641	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3834	. . . . . . . . . 10 ⊢ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐽)‘𝐴)
35		simpr 477	. . . . . . . . . 10 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3601	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 481	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3833	. . . . . . . . . 10 ⊢ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ⊆ 𝑦
39	38, 35	sseldi 3601	. . . . . . . . 9 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 20879	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑧 ∈ 𝑦)) → (𝑦 ∩ 𝐴) ≠ ∅)
41	30, 33, 36, 37, 39, 40	syl32anc 1334	. . . . . . . 8 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 1863	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 1108	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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) ∈ Conn) ∧ (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) ∈ Conn) ∧ (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 3641	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 20860	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
48	44, 46, 47	syl2anc 693	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3747	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3613	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3625	. . . . . . . . . 10 ⊢ 𝑋 ⊆ V
52		ssdif 3745	. . . . . . . . . 10 ⊢ (𝑋 ⊆ V → (𝑋 ∖ 𝐴) ⊆ (V ∖ 𝐴))
53	51, 52	ax-mp 5	. . . . . . . . 9 ⊢ (𝑋 ∖ 𝐴) ⊆ (V ∖ 𝐴)
54	50, 53	syl6ss 3615	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 4024	. . . . . . . 8 ⊢ (((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅ ↔ (𝑥 ∩ 𝑦) ⊆ (V ∖ 𝐴))
56	54, 55	sylibr 224	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 796	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 3613	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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	nconnsubb 21226	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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) ∈ Conn)
60	59	expr 643	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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) ∈ Conn))
61	1, 60	mt2d 131	. . . 4 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 450	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) ∧ (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 2971	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) → ∀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 1061	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) → 𝐽 ∈ (TopOn‘𝑋))
65	13	sseq2d 3633	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝐽))
66	65	biimpa 501	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ 𝐽)
67	22	clsss3 20863	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝐴) ⊆ ∪ 𝐽)
68	10, 67	sylan 488	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝐴) ⊆ ∪ 𝐽)
69	66, 68	syldan 487	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ ∪ 𝐽)
70	13	adantr 481	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 = ∪ 𝐽)
71	69, 70	sseqtr4d 3642	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
72	71	3adant3 1081	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
73		connsub 21224	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝐴) ⊆ 𝑋) → ((𝐽 ↾t ((cls‘𝐽)‘𝐴)) ∈ Conn ↔ ∀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 693	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) → ((𝐽 ↾t ((cls‘𝐽)‘𝐴)) ∈ Conn ↔ ∀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 247	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) → (𝐽 ↾t ((cls‘𝐽)‘𝐴)) ∈ Conn)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ∪ cuni 4436  ‘cfv 5888  (class class class)co 6650   ↾_t crest 16081  Topctop 20698  TopOnctopon 20715  clsccl 20822  Conncconn 21214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-conn 21215

This theorem is referenced by:  conncompcld  21237