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Theorem nrmsep 20971
 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 20950 . . . . . 6 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
21ad2antrr 758 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐽 ∈ Top)
3 elssuni 4403 . . . . . 6 (𝑥𝐽𝑥 𝐽)
43ad2antrl 760 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 𝐽)
5 eqid 2610 . . . . . 6 𝐽 = 𝐽
65clscld 20661 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
72, 4, 6syl2anc 691 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
85cldopn 20645 . . . 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 800 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐶𝑥)
11 incom 3767 . . . . 5 (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = (((cls‘𝐽)‘𝑥) ∩ 𝐷)
12 simprrr 801 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
1311, 12syl5eq 2656 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅)
14 simplr2 1097 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ∈ (Clsd‘𝐽))
155cldss 20643 . . . . 5 (𝐷 ∈ (Clsd‘𝐽) → 𝐷 𝐽)
16 reldisj 3972 . . . . 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 221 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)))
195sscls 20670 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
202, 4, 19syl2anc 691 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
21 ssrin 3800 . . . . 5 (𝑥 ⊆ ((cls‘𝐽)‘𝑥) → (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) ⊆ (((cls‘𝐽)‘𝑥) ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
2220, 21syl 17 . . . 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))))
23 disjdif 3992 . . . 4 (((cls‘𝐽)‘𝑥) ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅
24 sseq0 3927 . . . 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))) = ∅)
2522, 23, 24sylancl 693 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)
26 sseq2 3590 . . . . 5 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝐷𝑦𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
27 ineq2 3770 . . . . . 6 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝑥𝑦) = (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
2827eqeq1d 2612 . . . . 5 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝑥𝑦) = ∅ ↔ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅))
2926, 283anbi23d 1394 . . . 4 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅) ↔ (𝐶𝑥𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)))
3029rspcev 3282 . . 3 ((( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽 ∧ (𝐶𝑥𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)) → ∃𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
319, 10, 18, 25, 30syl13anc 1320 . 2 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ∃𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
32 nrmsep2 20970 . 2 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
3331, 32reximddv 3001 1 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
 Colors of variables: wff setvar class Syntax hints:   → 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  Nrmcnrm 20924 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-nrm 20931 This theorem is referenced by:  isnrm3  20973
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