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Theorem nrmsep2 17412
 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
Distinct variable groups:   ,   ,   ,

Proof of Theorem nrmsep2
StepHypRef Expression
1 simpl 444 . . 3
2 simpr2 964 . . . 4
3 eqid 2435 . . . . 5
43cldopn 17087 . . . 4
52, 4syl 16 . . 3
6 simpr1 963 . . 3
7 simpr3 965 . . . 4
83cldss 17085 . . . . 5
9 reldisj 3663 . . . . 5
106, 8, 93syl 19 . . . 4
117, 10mpbid 202 . . 3
12 nrmsep3 17411 . . 3
131, 5, 6, 11, 12syl13anc 1186 . 2
14 ssdifin0 3701 . . . 4
1514anim2i 553 . . 3
1615reximi 2805 . 2
1713, 16syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wrex 2698   cdif 3309   cin 3311   wss 3312  c0 3620  cuni 4007  cfv 5446  ccld 17072  ccl 17074  cnrm 17366 This theorem is referenced by:  nrmsep  17413  isnrm2  17414 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-top 16955  df-cld 17075  df-nrm 17373
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