Theorem clsconn 23659

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 1233	. . . . 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 1231	. . . . . . 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 1232	. . . . . . 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 789	. . . . . . 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 790	. . . . . . 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 1237	. . . . . . . . 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 4303	. . . . . . . . 9 ⊢ ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴)))
8	6, 7	sylib 221	. . . . . . . 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 486	. . . . . . . . . 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 23142	. . . . . . . . . 10 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
11	9, 10	syl 18	. . . . . . . . 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 486	. . . . . . . . . 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 23143	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
14	9, 13	syl 18	. . . . . . . . . 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 3970	. . . . . . . . 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		simpr 490	. . . . . . . . . 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)))
17	16	elin2d 4154	. . . . . . . . 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))
18	4	adantr 486	. . . . . . . . 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)
19	16	elin1d 4153	. . . . . . . . 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)
20		eqid 2762	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
21	20	clsndisj 23304	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑧 ∈ 𝑥)) → (𝑥 ∩ 𝐴) ≠ ∅)
22	11, 15, 17, 18, 19, 21	syl32anc 1405	. . . . . . . 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) ≠ ∅)
23	8, 22	exlimddv 1968	. . . . . . 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) ≠ ∅)
24		simprl2 1238	. . . . . . . . 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)) ≠ ∅)
25		n0 4303	. . . . . . . . 9 ⊢ ((𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴)))
26	24, 25	sylib 221	. . . . . . . 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)))
27	2	adantr 486	. . . . . . . . . 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))
28	27, 10	syl 18	. . . . . . . . 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)
29	3	adantr 486	. . . . . . . . . 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)
30	27, 13	syl 18	. . . . . . . . . 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)
31	29, 30	sseqtrd 3970	. . . . . . . . 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)
32		simpr 490	. . . . . . . . . 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)))
33	32	elin2d 4154	. . . . . . . . 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))
34	5	adantr 486	. . . . . . . . 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)
35	32	elin1d 4153	. . . . . . . . 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)
36	20	clsndisj 23304	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ (𝑦 ∈ 𝐽 ∧ 𝑧 ∈ 𝑦)) → (𝑦 ∩ 𝐴) ≠ ∅)
37	28, 31, 33, 34, 35, 36	syl32anc 1405	. . . . . . . 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) ≠ ∅)
38	26, 37	exlimddv 1968	. . . . . . 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) ≠ ∅)
39		simprl3 1239	. . . . . . . . . 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)))
40	2, 10	syl 18	. . . . . . . . . . . 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)
41	2, 13	syl 18	. . . . . . . . . . . . 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)
42	3, 41	sseqtrd 3970	. . . . . . . . . . . 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)
43	20	sscls 23285	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
44	40, 42, 43	syl2anc 596	. . . . . . . . . . 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))
45	44	sscond 4096	. . . . . . . . . 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))
46	39, 45	sstrd 3944	. . . . . . . . 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))
47		ssv 3958	. . . . . . . . . 10 ⊢ 𝑋 ⊆ V
48		ssdif 4094	. . . . . . . . . 10 ⊢ (𝑋 ⊆ V → (𝑋 ∖ 𝐴) ⊆ (V ∖ 𝐴))
49	47, 48	ax-mp 5	. . . . . . . . 9 ⊢ (𝑋 ∖ 𝐴) ⊆ (V ∖ 𝐴)
50	46, 49	sstrdi 3946	. . . . . . . 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))
51		disj2 4414	. . . . . . . 8 ⊢ (((𝑥 ∩ 𝑦) ∩ 𝐴) = ∅ ↔ (𝑥 ∩ 𝑦) ⊆ (V ∖ 𝐴))
52	50, 51	sylibr 237	. . . . . . 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) = ∅)
53		simprr 785	. . . . . . . 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))
54	44, 53	sstrd 3944	. . . . . . 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))
55	2, 3, 4, 5, 23, 38, 52, 54	nconnsubb 23652	. . . . . 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)
56	55	expr 462	. . . . 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))
57	1, 56	mt2d 137	. . . 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))
58	57	ex 418	. . 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)))
59	58	ralrimivva 3207	. 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)))
60		simp1 1154	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) → 𝐽 ∈ (TopOn‘𝑋))
61	13	sseq2d 3966	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ 𝐽))
62	61	biimpa 482	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ ∪ 𝐽)
63	20	clsss3 23288	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝐴) ⊆ ∪ 𝐽)
64	10, 62, 63	syl2an2r 698	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ ∪ 𝐽)
65	13	adantr 486	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → 𝑋 = ∪ 𝐽)
66	64, 65	sseqtrrd 3971	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
67	66	3adant3 1150	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
68		connsub 23650	. . 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))))
69	60, 67, 68	syl2anc 596	. 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))))
70	59, 69	mpbird 260	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ (𝐽 ↾t 𝐴) ∈ Conn) → (𝐽 ↾t ((cls‘𝐽)‘𝐴)) ∈ Conn)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  Vcvv 3453   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ∪ cuni 4870  ‘cfv 6537  (class class class)co 7416   ↾_t crest 17509  Topctop 23122  TopOnctopon 23139  clsccl 23247  Conncconn 23640

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-en 8956  df-fin 8959  df-fi 9384  df-rest 17511  df-topgen 17532  df-top 23123  df-topon 23140  df-bases 23175  df-cld 23248  df-ntr 23249  df-cls 23250  df-conn 23641

This theorem is used by:  conncompcld  23663