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Theorem nrmsep2 21958
 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 485 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐽 ∈ Nrm)
2 simpr2 1191 . . . 4 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐷 ∈ (Clsd‘𝐽))
3 eqid 2821 . . . . 5 𝐽 = 𝐽
43cldopn 21633 . . . 4 (𝐷 ∈ (Clsd‘𝐽) → ( 𝐽𝐷) ∈ 𝐽)
52, 4syl 17 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ( 𝐽𝐷) ∈ 𝐽)
6 simpr1 1190 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐶 ∈ (Clsd‘𝐽))
7 simpr3 1192 . . . 4 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → (𝐶𝐷) = ∅)
83cldss 21631 . . . . 5 (𝐶 ∈ (Clsd‘𝐽) → 𝐶 𝐽)
9 reldisj 4402 . . . . 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 234 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐶 ⊆ ( 𝐽𝐷))
12 nrmsep3 21957 . . 3 ((𝐽 ∈ Nrm ∧ (( 𝐽𝐷) ∈ 𝐽𝐶 ∈ (Clsd‘𝐽) ∧ 𝐶 ⊆ ( 𝐽𝐷))) → ∃𝑥𝐽 (𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)))
131, 5, 6, 11, 12syl13anc 1368 . 2 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)))
14 ssdifin0 4431 . . . 4 (((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
1514anim2i 618 . . 3 ((𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)) → (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
1615reximi 3243 . 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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∃wrex 3139   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  ∪ cuni 4832  ‘cfv 6350  Clsdccld 21618  clsccl 21620  Nrmcnrm 21912 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-iota 6309  df-fun 6352  df-fn 6353  df-fv 6358  df-top 21496  df-cld 21621  df-nrm 21919 This theorem is referenced by:  nrmsep  21959  isnrm2  21960
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