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Theorem regsep2 20990
 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 20947 . . . . . . 7 (𝐽 ∈ Reg → 𝐽 ∈ Top)
21ad2antrr 758 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐽 ∈ Top)
3 elssuni 4403 . . . . . . . 8 (𝑦𝐽𝑦 𝐽)
4 t1sep.1 . . . . . . . 8 𝑋 = 𝐽
53, 4syl6sseqr 3615 . . . . . . 7 (𝑦𝐽𝑦𝑋)
65ad2antrl 760 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝑦𝑋)
74clscld 20661 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑦𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
82, 6, 7syl2anc 691 . . . . 5 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
94cldopn 20645 . . . . 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 801 . . . . 5 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶))
124clsss3 20673 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑦𝑋) → ((cls‘𝐽)‘𝑦) ⊆ 𝑋)
132, 6, 12syl2anc 691 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((cls‘𝐽)‘𝑦) ⊆ 𝑋)
14 simplr1 1096 . . . . . . 7 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐶 ∈ (Clsd‘𝐽))
154cldss 20643 . . . . . . 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 3705 . . . . . 6 ((((cls‘𝐽)‘𝑦) ⊆ 𝑋𝐶𝑋) → (((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶) ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
1813, 16, 17syl2anc 691 . . . . 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 221 . . . 4 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)))
20 simprrl 800 . . . 4 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐴𝑦)
214sscls 20670 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑦𝑋) → 𝑦 ⊆ ((cls‘𝐽)‘𝑦))
222, 6, 21syl2anc 691 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝑦 ⊆ ((cls‘𝐽)‘𝑦))
23 sslin 3801 . . . . . 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 incom 3767 . . . . . 6 ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = (((cls‘𝐽)‘𝑦) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑦)))
26 disjdif 3992 . . . . . 6 (((cls‘𝐽)‘𝑦) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑦))) = ∅
2725, 26eqtri 2632 . . . . 5 ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = ∅
28 sseq0 3927 . . . . 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) = ∅)
2924, 27, 28sylancl 693 . . . 4 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)
30 sseq2 3590 . . . . . 6 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝐶𝑥𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
31 ineq1 3769 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝑥𝑦) = ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦))
3231eqeq1d 2612 . . . . . 6 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → ((𝑥𝑦) = ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅))
3330, 323anbi13d 1393 . . . . 5 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → ((𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅) ↔ (𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∧ 𝐴𝑦 ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)))
3433rspcev 3282 . . . 4 (((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∈ 𝐽 ∧ (𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∧ 𝐴𝑦 ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)) → ∃𝑥𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
3510, 19, 20, 29, 34syl13anc 1320 . . 3 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ∃𝑥𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
36 simpl 472 . . . 4 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐽 ∈ Reg)
37 simpr1 1060 . . . . 5 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐶 ∈ (Clsd‘𝐽))
384cldopn 20645 . . . . 5 (𝐶 ∈ (Clsd‘𝐽) → (𝑋𝐶) ∈ 𝐽)
3937, 38syl 17 . . . 4 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → (𝑋𝐶) ∈ 𝐽)
40 simpr2 1061 . . . . 5 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐴𝑋)
41 simpr3 1062 . . . . 5 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ¬ 𝐴𝐶)
4240, 41eldifd 3551 . . . 4 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐴 ∈ (𝑋𝐶))
43 regsep 20948 . . . 4 ((𝐽 ∈ Reg ∧ (𝑋𝐶) ∈ 𝐽𝐴 ∈ (𝑋𝐶)) → ∃𝑦𝐽 (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))
4436, 39, 42, 43syl3anc 1318 . . 3 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ∃𝑦𝐽 (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))
4535, 44reximddv 3001 . 2 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ∃𝑦𝐽𝑥𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
46 rexcom 3080 . 2 (∃𝑦𝐽𝑥𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅) ↔ ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
4745, 46sylib 207 1 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630  clsccl 20632  Regcreg 20923 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-cld 20633  df-cls 20635  df-reg 20930 This theorem is referenced by:  isreg2  20991
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