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Theorem nrmsep 21959
Description: In a normal space, disjoint closed sets are separated by open sets. (Contributed by Jeff Hankins, 1-Feb-2010.)
Assertion
Ref Expression
nrmsep ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦   𝑥,𝐽,𝑦

Proof of Theorem nrmsep
StepHypRef Expression
1 nrmtop 21938 . . . . . 6 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
21ad2antrr 724 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐽 ∈ Top)
3 elssuni 4860 . . . . . 6 (𝑥𝐽𝑥 𝐽)
43ad2antrl 726 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 𝐽)
5 eqid 2821 . . . . . 6 𝐽 = 𝐽
65clscld 21649 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
72, 4, 6syl2anc 586 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
85cldopn 21633 . . . 4 (((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽) → ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽)
97, 8syl 17 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽)
10 simprrl 779 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐶𝑥)
11 incom 4177 . . . . 5 (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = (((cls‘𝐽)‘𝑥) ∩ 𝐷)
12 simprrr 780 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
1311, 12syl5eq 2868 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅)
14 simplr2 1212 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ∈ (Clsd‘𝐽))
155cldss 21631 . . . . 5 (𝐷 ∈ (Clsd‘𝐽) → 𝐷 𝐽)
16 reldisj 4401 . . . . 5 (𝐷 𝐽 → ((𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅ ↔ 𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
1714, 15, 163syl 18 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ((𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅ ↔ 𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
1813, 17mpbid 234 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)))
195sscls 21658 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
202, 4, 19syl2anc 586 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
2120ssrind 4211 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) ⊆ (((cls‘𝐽)‘𝑥) ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
22 disjdif 4420 . . . 4 (((cls‘𝐽)‘𝑥) ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅
23 sseq0 4352 . . . 4 (((𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) ⊆ (((cls‘𝐽)‘𝑥) ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) ∧ (((cls‘𝐽)‘𝑥) ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅) → (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)
2421, 22, 23sylancl 588 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)
25 sseq2 3992 . . . . 5 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝐷𝑦𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
26 ineq2 4182 . . . . . 6 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝑥𝑦) = (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
2726eqeq1d 2823 . . . . 5 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝑥𝑦) = ∅ ↔ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅))
2825, 273anbi23d 1435 . . . 4 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅) ↔ (𝐶𝑥𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)))
2928rspcev 3622 . . 3 ((( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽 ∧ (𝐶𝑥𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)) → ∃𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
309, 10, 18, 24, 29syl13anc 1368 . 2 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ∃𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
31 nrmsep2 21958 . 2 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
3230, 31reximddv 3275 1 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wrex 3139  cdif 3932  cin 3934  wss 3935  c0 4290   cuni 4831  cfv 6349  Topctop 21495  Clsdccld 21618  clsccl 21620  Nrmcnrm 21912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-top 21496  df-cld 21621  df-cls 21623  df-nrm 21919
This theorem is referenced by:  isnrm3  21961
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