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Theorem clsconn 22030
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 1209 . . . . 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 1207 . . . . . . 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 1208 . . . . . . 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 775 . . . . . . 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 776 . . . . . . 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 1213 . . . . . . . . 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 4308 . . . . . . . . 9 ((𝑥 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑥 ∩ ((cls‘𝐽)‘𝐴)))
86, 7sylib 220 . . . . . . . 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 483 . . . . . . . . . 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 21513 . . . . . . . . . 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 483 . . . . . . . . . 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 21514 . . . . . . . . . . 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 4005 . . . . . . . . 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 simpr 487 . . . . . . . . . 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)))
1716elin2d 4174 . . . . . . . . 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))
184adantr 483 . . . . . . . . 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)
1916elin1d 4173 . . . . . . . . 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)
20 eqid 2819 . . . . . . . . . 10 𝐽 = 𝐽
2120clsndisj 21675 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴 𝐽𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ (𝑥𝐽𝑧𝑥)) → (𝑥𝐴) ≠ ∅)
2211, 15, 17, 18, 19, 21syl32anc 1373 . . . . . . . 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) ≠ ∅)
238, 22exlimddv 1930 . . . . . . 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) ≠ ∅)
24 simprl2 1214 . . . . . . . . 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)) ≠ ∅)
25 n0 4308 . . . . . . . . 9 ((𝑦 ∩ ((cls‘𝐽)‘𝐴)) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝑦 ∩ ((cls‘𝐽)‘𝐴)))
2624, 25sylib 220 . . . . . . . 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)))
272adantr 483 . . . . . . . . . 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))
2827, 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)
293adantr 483 . . . . . . . . . 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)
3027, 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)
3129, 30sseqtrd 4005 . . . . . . . . 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)
32 simpr 487 . . . . . . . . . 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)))
3332elin2d 4174 . . . . . . . . 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))
345adantr 483 . . . . . . . . 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)
3532elin1d 4173 . . . . . . . . 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)
3620clsndisj 21675 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐴 𝐽𝑧 ∈ ((cls‘𝐽)‘𝐴)) ∧ (𝑦𝐽𝑧𝑦)) → (𝑦𝐴) ≠ ∅)
3728, 31, 33, 34, 35, 36syl32anc 1373 . . . . . . . 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) ≠ ∅)
3826, 37exlimddv 1930 . . . . . . 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) ≠ ∅)
39 simprl3 1215 . . . . . . . . . 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)))
402, 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)
412, 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)
423, 41sseqtrd 4005 . . . . . . . . . . . 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)
4320sscls 21656 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴))
4440, 42, 43syl2anc 586 . . . . . . . . . . 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))
4544sscond 4116 . . . . . . . . . 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))
4639, 45sstrd 3975 . . . . . . . . 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))
47 ssv 3989 . . . . . . . . . 10 𝑋 ⊆ V
48 ssdif 4114 . . . . . . . . . 10 (𝑋 ⊆ V → (𝑋𝐴) ⊆ (V ∖ 𝐴))
4947, 48ax-mp 5 . . . . . . . . 9 (𝑋𝐴) ⊆ (V ∖ 𝐴)
5046, 49sstrdi 3977 . . . . . . . 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))
51 disj2 4405 . . . . . . . 8 (((𝑥𝑦) ∩ 𝐴) = ∅ ↔ (𝑥𝑦) ⊆ (V ∖ 𝐴))
5250, 51sylibr 236 . . . . . . 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) = ∅)
53 simprr 771 . . . . . . . 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))
5444, 53sstrd 3975 . . . . . . 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))
552, 3, 4, 5, 23, 38, 52, 54nconnsubb 22023 . . . . . 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)
5655expr 459 . . . . 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))
571, 56mt2d 138 . . . 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))
5857ex 415 . . 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)))
5958ralrimivva 3189 . 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)))
60 simp1 1131 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) → 𝐽 ∈ (TopOn‘𝑋))
6113sseq2d 3997 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → (𝐴𝑋𝐴 𝐽))
6261biimpa 479 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 𝐽)
6320clsss3 21659 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → ((cls‘𝐽)‘𝐴) ⊆ 𝐽)
6410, 62, 63syl2an2r 683 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝐽)
6513adantr 483 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑋 = 𝐽)
6664, 65sseqtrrd 4006 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
67663adant3 1127 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋 ∧ (𝐽t 𝐴) ∈ Conn) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
68 connsub 22021 . . 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))))
6960, 67, 68syl2anc 586 . 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))))
7059, 69mpbird 259 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 208  wa 398  w3a 1082   = wceq 1531  wex 1774  wcel 2108  wne 3014  wral 3136  Vcvv 3493  cdif 3931  cun 3932  cin 3933  wss 3934  c0 4289   cuni 4830  cfv 6348  (class class class)co 7148  t crest 16686  Topctop 21493  TopOnctopon 21510  clsccl 21618  Conncconn 22011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-oadd 8098  df-er 8281  df-en 8502  df-fin 8505  df-fi 8867  df-rest 16688  df-topgen 16709  df-top 21494  df-topon 21511  df-bases 21546  df-cld 21619  df-ntr 21620  df-cls 21621  df-conn 22012
This theorem is referenced by:  conncompcld  22034
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