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Theorem nrmsep2 23364
Description: In a normal space, any two disjoint closed sets have the property that each one is a subset of an open set whose closure is disjoint from the other. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
nrmsep2 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐷   𝑥,𝐽

Proof of Theorem nrmsep2
StepHypRef Expression
1 simpl 482 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐽 ∈ Nrm)
2 simpr2 1196 . . . 4 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐷 ∈ (Clsd‘𝐽))
3 eqid 2737 . . . . 5 𝐽 = 𝐽
43cldopn 23039 . . . 4 (𝐷 ∈ (Clsd‘𝐽) → ( 𝐽𝐷) ∈ 𝐽)
52, 4syl 17 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ( 𝐽𝐷) ∈ 𝐽)
6 simpr1 1195 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐶 ∈ (Clsd‘𝐽))
7 simpr3 1197 . . . 4 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → (𝐶𝐷) = ∅)
83cldss 23037 . . . . 5 (𝐶 ∈ (Clsd‘𝐽) → 𝐶 𝐽)
9 reldisj 4453 . . . . 5 (𝐶 𝐽 → ((𝐶𝐷) = ∅ ↔ 𝐶 ⊆ ( 𝐽𝐷)))
106, 8, 93syl 18 . . . 4 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ((𝐶𝐷) = ∅ ↔ 𝐶 ⊆ ( 𝐽𝐷)))
117, 10mpbid 232 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐶 ⊆ ( 𝐽𝐷))
12 nrmsep3 23363 . . 3 ((𝐽 ∈ Nrm ∧ (( 𝐽𝐷) ∈ 𝐽𝐶 ∈ (Clsd‘𝐽) ∧ 𝐶 ⊆ ( 𝐽𝐷))) → ∃𝑥𝐽 (𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)))
131, 5, 6, 11, 12syl13anc 1374 . 2 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)))
14 ssdifin0 4486 . . . 4 (((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
1514anim2i 617 . . 3 ((𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)) → (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
1615reximi 3084 . 2 (∃𝑥𝐽 (𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)) → ∃𝑥𝐽 (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
1713, 16syl 17 1 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
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  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-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-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  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-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-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-top 22900  df-cld 23027  df-nrm 23325
This theorem is referenced by:  nrmsep  23365  isnrm2  23366
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