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Theorem nrmsep 23365
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 23344 . . . . . 6 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
21ad2antrr 726 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐽 ∈ Top)
3 elssuni 4937 . . . . . 6 (𝑥𝐽𝑥 𝐽)
43ad2antrl 728 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 𝐽)
5 eqid 2737 . . . . . 6 𝐽 = 𝐽
65clscld 23055 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
72, 4, 6syl2anc 584 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ((cls‘𝐽)‘𝑥) ∈ (Clsd‘𝐽))
85cldopn 23039 . . . 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 781 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐶𝑥)
11 incom 4209 . . . . 5 (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = (((cls‘𝐽)‘𝑥) ∩ 𝐷)
12 simprrr 782 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
1311, 12eqtrid 2789 . . . 4 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝐷 ∩ ((cls‘𝐽)‘𝑥)) = ∅)
14 simplr2 1217 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ∈ (Clsd‘𝐽))
155cldss 23037 . . . . 5 (𝐷 ∈ (Clsd‘𝐽) → 𝐷 𝐽)
16 reldisj 4453 . . . . 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 232 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)))
195sscls 23064 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
202, 4, 19syl2anc 584 . . . . 5 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → 𝑥 ⊆ ((cls‘𝐽)‘𝑥))
2120ssrind 4244 . . . 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 4472 . . . 4 (((cls‘𝐽)‘𝑥) ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅
23 sseq0 4403 . . . 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 586 . . 3 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅)
25 sseq2 4010 . . . . 5 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝐷𝑦𝐷 ⊆ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
26 ineq2 4214 . . . . . 6 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → (𝑥𝑦) = (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))))
2726eqeq1d 2739 . . . . 5 (𝑦 = ( 𝐽 ∖ ((cls‘𝐽)‘𝑥)) → ((𝑥𝑦) = ∅ ↔ (𝑥 ∩ ( 𝐽 ∖ ((cls‘𝐽)‘𝑥))) = ∅))
2825, 273anbi23d 1441 . . . 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 1374 . 2 (((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) ∧ (𝑥𝐽 ∧ (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))) → ∃𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
31 nrmsep2 23364 . 2 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
3230, 31reximddv 3171 1 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽𝑦𝐽 (𝐶𝑥𝐷𝑦 ∧ (𝑥𝑦) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  cdif 3948  cin 3950  wss 3951  c0 4333   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024  clsccl 23026  Nrmcnrm 23318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-cld 23027  df-cls 23029  df-nrm 23325
This theorem is referenced by:  isnrm3  23367
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