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Theorem nrmsep 21355
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 21334 . . . . . 6 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
21ad2antrr 764 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐽 ∈ Top)
3 elssuni 4611 . . . . . 6 (𝑥𝐽𝑥 𝐽)
43ad2antrl 766 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 𝐽)
5 eqid 2752 . . . . . 6 𝐽 = 𝐽
65clscld 21045 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
72, 4, 6syl2anc 696 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
85cldopn 21029 . . . 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 823 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐶𝑥)
11 incom 3940 . . . . 5 (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = (((cls‘𝐽)‘𝑥) ∩ 𝐷)
12 simprrr 824 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
1311, 12syl5eq 2798 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅)
14 simplr2 1260 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ∈ (Clsd‘𝐽))
155cldss 21027 . . . . 5 (𝐷 ∈ (Clsd‘𝐽) → 𝐷 𝐽)
16 reldisj 4155 . . . . 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 222 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)))
195sscls 21054 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
202, 4, 19syl2anc 696 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
21 ssrin 3973 . . . . 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 4176 . . . 4 (((cls‘𝐽)‘𝑥) ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅
24 sseq0 4110 . . . 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 697 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)
26 sseq2 3760 . . . . 5 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝐷𝑦𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
27 ineq2 3943 . . . . . 6 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝑥𝑦) = (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
2827eqeq1d 2754 . . . . 5 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝑥𝑦) = ∅ ↔ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅))
2926, 283anbi23d 1543 . . . 4 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅) ↔ (𝐶𝑥𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)))
3029rspcev 3441 . . 3 ((( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽 ∧ (𝐶𝑥𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)) → ∃𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
319, 10, 18, 25, 30syl13anc 1475 . 2 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ∃𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
32 nrmsep2 21354 . 2 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
3331, 32reximddv 3148 1 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wrex 3043  cdif 3704  cin 3706  wss 3707  c0 4050   cuni 4580  cfv 6041  Topctop 20892  Clsdccld 21014  clsccl 21016  Nrmcnrm 21308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-top 20893  df-cld 21017  df-cls 21019  df-nrm 21315
This theorem is referenced by:  isnrm3  21357
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