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Theorem clsconn 21454
Description: The closure of a connected set is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
clsconn ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) → (𝐽t ((cls‘𝐽)‘𝐴)) ∈ Conn)

Proof of Theorem clsconn
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll3 1258 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴)))) → (𝐽t 𝐴) ∈ Conn)
2 simpll1 1254 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → 𝐽 ∈ (TopOn‘𝑋))
3 simpll2 1256 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → 𝐴𝑋)
4 simplrl 762 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → 𝑥𝐽)
5 simplrr 763 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → 𝑦𝐽)
6 simprl1 1266 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → (𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅)
7 n0 4078 . . . . . . . . 9 ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴)))
86, 7sylib 208 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → ∃𝑧 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴)))
92adantr 466 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ (TopOn‘𝑋))
10 topontop 20938 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
119, 10syl 17 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ Top)
123adantr 466 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝐴𝑋)
13 toponuni 20939 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
149, 13syl 17 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝑋 = 𝐽)
1512, 14sseqtrd 3790 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝐴 𝐽)
16 inss2 3982 . . . . . . . . . 10 (𝑥 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐽)‘𝐴)
17 simpr 471 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴)))
1816, 17sseldi 3750 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧 ∈ ((cls‘𝐽)‘𝐴))
194adantr 466 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝑥𝐽)
20 inss1 3981 . . . . . . . . . 10 (𝑥 ∩ ((cls‘𝐽)‘𝐴)) ⊆ 𝑥
2120, 17sseldi 3750 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧𝑥)
22 eqid 2771 . . . . . . . . . 10 𝐽 = 𝐽
2322clsndisj 21100 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴 𝐽𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ (𝑥𝐽𝑧𝑥)) → (𝑥𝐴) ≠ ∅)
2411, 15, 18, 19, 21, 23syl32anc 1484 . . . . . . . 8 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴))) → (𝑥𝐴) ≠ ∅)
258, 24exlimddv 2015 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → (𝑥𝐴) ≠ ∅)
26 simprl2 1268 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅)
27 n0 4078 . . . . . . . . 9 ((𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴)))
2826, 27sylib 208 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → ∃𝑧 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴)))
292adantr 466 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ (TopOn‘𝑋))
3029, 10syl 17 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝐽 ∈ Top)
313adantr 466 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝐴𝑋)
3229, 13syl 17 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝑋 = 𝐽)
3331, 32sseqtrd 3790 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝐴 𝐽)
34 inss2 3982 . . . . . . . . . 10 (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ⊆ ((cls‘𝐽)‘𝐴)
35 simpr 471 . . . . . . . . . 10 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴)))
3634, 35sseldi 3750 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧 ∈ ((cls‘𝐽)‘𝐴))
375adantr 466 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝑦𝐽)
38 inss1 3981 . . . . . . . . . 10 (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ⊆ 𝑦
3938, 35sseldi 3750 . . . . . . . . 9 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → 𝑧𝑦)
4022clsndisj 21100 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴 𝐽𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ (𝑦𝐽𝑧𝑦)) → (𝑦𝐴) ≠ ∅)
4130, 33, 36, 37, 39, 40syl32anc 1484 . . . . . . . 8 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) ∧ 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴))) → (𝑦𝐴) ≠ ∅)
4228, 41exlimddv 2015 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → (𝑦𝐴) ≠ ∅)
43 simprl3 1270 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴)))
442, 10syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → 𝐽 ∈ Top)
452, 13syl 17 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → 𝑋 = 𝐽)
463, 45sseqtrd 3790 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → 𝐴 𝐽)
4722sscls 21081 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
4844, 46, 47syl2anc 573 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
4948sscond 3898 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → (𝑋 ∖ ((cls‘𝐽)‘𝐴)) ⊆ (𝑋𝐴))
5043, 49sstrd 3762 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → (𝑥𝑦) ⊆ (𝑋𝐴))
51 ssv 3774 . . . . . . . . . 10 𝑋 ⊆ V
52 ssdif 3896 . . . . . . . . . 10 (𝑋 ⊆ V → (𝑋𝐴) ⊆ (V ∖ 𝐴))
5351, 52ax-mp 5 . . . . . . . . 9 (𝑋𝐴) ⊆ (V ∖ 𝐴)
5450, 53syl6ss 3764 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → (𝑥𝑦) ⊆ (V ∖ 𝐴))
55 disj2 4168 . . . . . . . 8 (((𝑥𝑦) ∩ 𝐴) = ∅ ↔ (𝑥𝑦) ⊆ (V ∖ 𝐴))
5654, 55sylibr 224 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → ((𝑥𝑦) ∩ 𝐴) = ∅)
57 simprr 756 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))
5848, 57sstrd 3762 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → 𝐴 ⊆ (𝑥𝑦))
592, 3, 4, 5, 25, 42, 56, 58nconnsubb 21447 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) ∧ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))) → ¬ (𝐽t 𝐴) ∈ Conn)
6059expr 444 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴)))) → (((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦) → ¬ (𝐽t 𝐴) ∈ Conn))
611, 60mt2d 133 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) ∧ ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴)))) → ¬ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))
6261ex 397 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) ∧ (𝑥𝐽𝑦𝐽)) → (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) → ¬ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦)))
6362ralrimivva 3120 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) → ∀𝑥𝐽𝑦𝐽 (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) → ¬ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦)))
64 simp1 1130 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) → 𝐽 ∈ (TopOn‘𝑋))
6513sseq2d 3782 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → (𝐴𝑋𝐴 𝐽))
6665biimpa 462 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 𝐽)
6722clsss3 21084 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → ((cls‘𝐽)‘𝐴) ⊆ 𝐽)
6810, 67sylan 569 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 𝐽) → ((cls‘𝐽)‘𝐴) ⊆ 𝐽)
6966, 68syldan 579 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝐽)
7013adantr 466 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 = 𝐽)
7169, 70sseqtr4d 3791 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
72713adant3 1126 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
73 connsub 21445 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝐴) ⊆ 𝑋) → ((𝐽t ((cls‘𝐽)‘𝐴)) ∈ Conn ↔ ∀𝑥𝐽𝑦𝐽 (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) → ¬ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))))
7464, 72, 73syl2anc 573 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) → ((𝐽t ((cls‘𝐽)‘𝐴)) ∈ Conn ↔ ∀𝑥𝐽𝑦𝐽 (((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ∧ (𝑥𝑦) ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝐴))) → ¬ ((cls‘𝐽)‘𝐴) ⊆ (𝑥𝑦))))
7563, 74mpbird 247 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) → (𝐽t ((cls‘𝐽)‘𝐴)) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063   cuni 4574  cfv 6031  (class class class)co 6793  t crest 16289  Topctop 20918  TopOnctopon 20935  clsccl 21043  Conncconn 21435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-er 7896  df-en 8110  df-fin 8113  df-fi 8473  df-rest 16291  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-cld 21044  df-ntr 21045  df-cls 21046  df-conn 21436
This theorem is referenced by:  conncompcld  21458
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