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Theorem nrmsep2 21892
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 483 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐽 ∈ Nrm)
2 simpr2 1187 . . . 4 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐷 ∈ (Clsd‘𝐽))
3 eqid 2818 . . . . 5 𝐽 = 𝐽
43cldopn 21567 . . . 4 (𝐷 ∈ (Clsd‘𝐽) → ( 𝐽𝐷) ∈ 𝐽)
52, 4syl 17 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ( 𝐽𝐷) ∈ 𝐽)
6 simpr1 1186 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐶 ∈ (Clsd‘𝐽))
7 simpr3 1188 . . . 4 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → (𝐶𝐷) = ∅)
83cldss 21565 . . . . 5 (𝐶 ∈ (Clsd‘𝐽) → 𝐶 𝐽)
9 reldisj 4398 . . . . 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 233 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐶 ⊆ ( 𝐽𝐷))
12 nrmsep3 21891 . . 3 ((𝐽 ∈ Nrm ∧ (( 𝐽𝐷) ∈ 𝐽𝐶 ∈ (Clsd‘𝐽) ∧ 𝐶 ⊆ ( 𝐽𝐷))) → ∃𝑥𝐽 (𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)))
131, 5, 6, 11, 12syl13anc 1364 . 2 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)))
14 ssdifin0 4427 . . . 4 (((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
1514anim2i 616 . . 3 ((𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)) → (𝐶𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅))
1615reximi 3240 . 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  cdif 3930  cin 3932  wss 3933  c0 4288   cuni 4830  cfv 6348  Clsdccld 21552  clsccl 21554  Nrmcnrm 21846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-top 21430  df-cld 21555  df-nrm 21853
This theorem is referenced by:  nrmsep  21893  isnrm2  21894
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