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Theorem regsep2 23100
Description: In a regular space, a closed set is separated by open sets from a point not in it. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
t1sep.1 𝑋 = 𝐽
Assertion
Ref Expression
regsep2 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐶,𝑦   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦

Proof of Theorem regsep2
StepHypRef Expression
1 regtop 23057 . . . . . . 7 (𝐽 ∈ Reg → 𝐽 ∈ Top)
21ad2antrr 722 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐽 ∈ Top)
3 elssuni 4940 . . . . . . . 8 (𝑦𝐽𝑦 𝐽)
4 t1sep.1 . . . . . . . 8 𝑋 = 𝐽
53, 4sseqtrrdi 4032 . . . . . . 7 (𝑦𝐽𝑦𝑋)
65ad2antrl 724 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝑦𝑋)
74clscld 22771 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑦𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
82, 6, 7syl2anc 582 . . . . 5 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
94cldopn 22755 . . . . 5 (((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽) → (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∈ 𝐽)
108, 9syl 17 . . . 4 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∈ 𝐽)
11 simprrr 778 . . . . 5 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶))
124clsss3 22783 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑦𝑋) → ((cls‘𝐽)‘𝑦) ⊆ 𝑋)
132, 6, 12syl2anc 582 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((cls‘𝐽)‘𝑦) ⊆ 𝑋)
14 simplr1 1213 . . . . . . 7 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐶 ∈ (Clsd‘𝐽))
154cldss 22753 . . . . . . 7 (𝐶 ∈ (Clsd‘𝐽) → 𝐶𝑋)
1614, 15syl 17 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐶𝑋)
17 ssconb 4136 . . . . . 6 ((((cls‘𝐽)‘𝑦) ⊆ 𝑋𝐶𝑋) → (((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶) ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
1813, 16, 17syl2anc 582 . . . . 5 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → (((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶) ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
1911, 18mpbid 231 . . . 4 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)))
20 simprrl 777 . . . 4 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐴𝑦)
214sscls 22780 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑦𝑋) → 𝑦 ⊆ ((cls‘𝐽)‘𝑦))
222, 6, 21syl2anc 582 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝑦 ⊆ ((cls‘𝐽)‘𝑦))
23 sslin 4233 . . . . . 6 (𝑦 ⊆ ((cls‘𝐽)‘𝑦) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) ⊆ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)))
2422, 23syl 17 . . . . 5 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) ⊆ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)))
25 disjdifr 4471 . . . . 5 ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = ∅
26 sseq0 4398 . . . . 5 ((((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) ⊆ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = ∅) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)
2724, 25, 26sylancl 584 . . . 4 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)
28 sseq2 4007 . . . . . 6 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝐶𝑥𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
29 ineq1 4204 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝑥𝑦) = ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦))
3029eqeq1d 2732 . . . . . 6 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → ((𝑥𝑦) = ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅))
3128, 303anbi13d 1436 . . . . 5 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → ((𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅) ↔ (𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∧ 𝐴𝑦 ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)))
3231rspcev 3611 . . . 4 (((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∈ 𝐽 ∧ (𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∧ 𝐴𝑦 ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)) → ∃𝑥𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
3310, 19, 20, 27, 32syl13anc 1370 . . 3 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ∃𝑥𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
34 simpl 481 . . . 4 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐽 ∈ Reg)
35 simpr1 1192 . . . . 5 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐶 ∈ (Clsd‘𝐽))
364cldopn 22755 . . . . 5 (𝐶 ∈ (Clsd‘𝐽) → (𝑋𝐶) ∈ 𝐽)
3735, 36syl 17 . . . 4 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → (𝑋𝐶) ∈ 𝐽)
38 simpr2 1193 . . . . 5 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐴𝑋)
39 simpr3 1194 . . . . 5 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ¬ 𝐴𝐶)
4038, 39eldifd 3958 . . . 4 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐴 ∈ (𝑋𝐶))
41 regsep 23058 . . . 4 ((𝐽 ∈ Reg ∧ (𝑋𝐶) ∈ 𝐽𝐴 ∈ (𝑋𝐶)) → ∃𝑦𝐽 (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))
4234, 37, 40, 41syl3anc 1369 . . 3 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ∃𝑦𝐽 (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))
4333, 42reximddv 3169 . 2 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ∃𝑦𝐽𝑥𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
44 rexcom 3285 . 2 (∃𝑦𝐽𝑥𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅) ↔ ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
4543, 44sylib 217 1 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1085   = wceq 1539  wcel 2104  wrex 3068  cdif 3944  cin 3946  wss 3947  c0 4321   cuni 4907  cfv 6542  Topctop 22615  Clsdccld 22740  clsccl 22742  Regcreg 23033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-top 22616  df-cld 22743  df-cls 22745  df-reg 23040
This theorem is referenced by:  isreg2  23101
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