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Theorem nrmsep 21490
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 21469 . . . . . 6 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
21ad2antrr 718 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐽 ∈ Top)
3 elssuni 4659 . . . . . 6 (𝑥𝐽𝑥 𝐽)
43ad2antrl 720 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 𝐽)
5 eqid 2799 . . . . . 6 𝐽 = 𝐽
65clscld 21180 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
72, 4, 6syl2anc 580 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
85cldopn 21164 . . . 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 4003 . . . . 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 2845 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅)
14 simplr2 1278 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ∈ (Clsd‘𝐽))
155cldss 21162 . . . . 5 (𝐷 ∈ (Clsd‘𝐽) → 𝐷 𝐽)
16 reldisj 4215 . . . . 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 224 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)))
195sscls 21189 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
202, 4, 19syl2anc 580 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
2120ssrind 4035 . . . 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 4234 . . . 4 (((cls‘𝐽)‘𝑥) ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅
23 sseq0 4171 . . . 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 581 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)
25 sseq2 3823 . . . . 5 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝐷𝑦𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
26 ineq2 4006 . . . . . 6 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝑥𝑦) = (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
2726eqeq1d 2801 . . . . 5 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝑥𝑦) = ∅ ↔ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅))
2825, 273anbi23d 1564 . . . 4 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅) ↔ (𝐶𝑥𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)))
2928rspcev 3497 . . 3 ((( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∈ 𝐽 ∧ (𝐶𝑥𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) ∧ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)) → ∃𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
309, 10, 18, 24, 29syl13anc 1492 . 2 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ∃𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
31 nrmsep2 21489 . 2 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
3230, 31reximddv 3198 1 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wrex 3090  cdif 3766  cin 3768  wss 3769  c0 4115   cuni 4628  cfv 6101  Topctop 21026  Clsdccld 21149  clsccl 21151  Nrmcnrm 21443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-iin 4713  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-top 21027  df-cld 21152  df-cls 21154  df-nrm 21450
This theorem is referenced by:  isnrm3  21492
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