MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  regsep2 Structured version   Visualization version   GIF version

Theorem regsep2 21382
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 21339 . . . . . . 7 (𝐽 ∈ Reg → 𝐽 ∈ Top)
21ad2antrr 764 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐽 ∈ Top)
3 elssuni 4619 . . . . . . . 8 (𝑦𝐽𝑦 𝐽)
4 t1sep.1 . . . . . . . 8 𝑋 = 𝐽
53, 4syl6sseqr 3793 . . . . . . 7 (𝑦𝐽𝑦𝑋)
65ad2antrl 766 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝑦𝑋)
74clscld 21053 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑦𝑋) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
82, 6, 7syl2anc 696 . . . . 5 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((cls‘𝐽)‘𝑦) ∈ (Clsd‘𝐽))
94cldopn 21037 . . . . 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 824 . . . . 5 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶))
124clsss3 21065 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑦𝑋) → ((cls‘𝐽)‘𝑦) ⊆ 𝑋)
132, 6, 12syl2anc 696 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((cls‘𝐽)‘𝑦) ⊆ 𝑋)
14 simplr1 1261 . . . . . . 7 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐶 ∈ (Clsd‘𝐽))
154cldss 21035 . . . . . . 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 3886 . . . . . 6 ((((cls‘𝐽)‘𝑦) ⊆ 𝑋𝐶𝑋) → (((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶) ↔ 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
1813, 16, 17syl2anc 696 . . . . 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 222 . . . 4 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)))
20 simprrl 823 . . . 4 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝐴𝑦)
214sscls 21062 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑦𝑋) → 𝑦 ⊆ ((cls‘𝐽)‘𝑦))
222, 6, 21syl2anc 696 . . . . . 6 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → 𝑦 ⊆ ((cls‘𝐽)‘𝑦))
23 sslin 3982 . . . . . 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 3948 . . . . . 6 ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = (((cls‘𝐽)‘𝑦) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑦)))
26 disjdif 4184 . . . . . 6 (((cls‘𝐽)‘𝑦) ∩ (𝑋 ∖ ((cls‘𝐽)‘𝑦))) = ∅
2725, 26eqtri 2782 . . . . 5 ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ ((cls‘𝐽)‘𝑦)) = ∅
28 sseq0 4118 . . . . 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 697 . . . 4 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)
30 sseq2 3768 . . . . . 6 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝐶𝑥𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦))))
31 ineq1 3950 . . . . . . 7 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → (𝑥𝑦) = ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦))
3231eqeq1d 2762 . . . . . 6 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → ((𝑥𝑦) = ∅ ↔ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅))
3330, 323anbi13d 1550 . . . . 5 (𝑥 = (𝑋 ∖ ((cls‘𝐽)‘𝑦)) → ((𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅) ↔ (𝐶 ⊆ (𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∧ 𝐴𝑦 ∧ ((𝑋 ∖ ((cls‘𝐽)‘𝑦)) ∩ 𝑦) = ∅)))
3433rspcev 3449 . . . 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 1479 . . 3 (((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) ∧ (𝑦𝐽 ∧ (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))) → ∃𝑥𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
36 simpl 474 . . . 4 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐽 ∈ Reg)
37 simpr1 1234 . . . . 5 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐶 ∈ (Clsd‘𝐽))
384cldopn 21037 . . . . 5 (𝐶 ∈ (Clsd‘𝐽) → (𝑋𝐶) ∈ 𝐽)
3937, 38syl 17 . . . 4 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → (𝑋𝐶) ∈ 𝐽)
40 simpr2 1236 . . . . 5 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐴𝑋)
41 simpr3 1238 . . . . 5 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ¬ 𝐴𝐶)
4240, 41eldifd 3726 . . . 4 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → 𝐴 ∈ (𝑋𝐶))
43 regsep 21340 . . . 4 ((𝐽 ∈ Reg ∧ (𝑋𝐶) ∈ 𝐽𝐴 ∈ (𝑋𝐶)) → ∃𝑦𝐽 (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))
4436, 39, 42, 43syl3anc 1477 . . 3 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ∃𝑦𝐽 (𝐴𝑦 ∧ ((cls‘𝐽)‘𝑦) ⊆ (𝑋𝐶)))
4535, 44reximddv 3156 . 2 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ∃𝑦𝐽𝑥𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
46 rexcom 3237 . 2 (∃𝑦𝐽𝑥𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅) ↔ ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
4745, 46sylib 208 1 ((𝐽 ∈ Reg ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐴𝑋 ∧ ¬ 𝐴𝐶)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐴𝑦 ∧ (𝑥𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wrex 3051  cdif 3712  cin 3714  wss 3715  c0 4058   cuni 4588  cfv 6049  Topctop 20900  Clsdccld 21022  clsccl 21024  Regcreg 21315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20901  df-cld 21025  df-cls 21027  df-reg 21322
This theorem is referenced by:  isreg2  21383
  Copyright terms: Public domain W3C validator