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Theorem nrmsep2 22730
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 484 . . 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 2733 . . . . 5 𝐽 = 𝐽
43cldopn 22405 . . . 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 22403 . . . . 5 (𝐶 ∈ (Clsd‘𝐽) → 𝐶 𝐽)
9 reldisj 4415 . . . . 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 231 . . 3 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → 𝐶 ⊆ ( 𝐽𝐷))
12 nrmsep3 22729 . . 3 ((𝐽 ∈ Nrm ∧ (( 𝐽𝐷) ∈ 𝐽𝐶 ∈ (Clsd‘𝐽) ∧ 𝐶 ⊆ ( 𝐽𝐷))) → ∃𝑥𝐽 (𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)))
131, 5, 6, 11, 12syl13anc 1373 . 2 ((𝐽 ∈ Nrm ∧ (𝐶 ∈ (Clsd‘𝐽) ∧ 𝐷 ∈ (Clsd‘𝐽) ∧ (𝐶𝐷) = ∅)) → ∃𝑥𝐽 (𝐶𝑥 ∧ ((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷)))
14 ssdifin0 4447 . . . 4 (((cls‘𝐽)‘𝑥) ⊆ ( 𝐽𝐷) → (((cls‘𝐽)‘𝑥) ∩ 𝐷) = ∅)
1514anim2i 618 . . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3070  cdif 3911  cin 3913  wss 3914  c0 4286   cuni 4869  cfv 6500  Clsdccld 22390  clsccl 22392  Nrmcnrm 22684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-top 22266  df-cld 22393  df-nrm 22691
This theorem is referenced by:  nrmsep  22731  isnrm2  22732
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