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Theorem nrmsep3 17424
 Description: In a normal space, given a closed set inside an open set , there is an open set such that . (Contributed by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
nrmsep3
Distinct variable groups:   ,   ,   ,

Proof of Theorem nrmsep3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnrm 17404 . . . . . 6
21simprbi 452 . . . . 5
3 pweq 3804 . . . . . . . 8
43ineq2d 3544 . . . . . . 7
5 sseq2 3372 . . . . . . . . 9
65anbi2d 686 . . . . . . . 8
76rexbidv 2728 . . . . . . 7
84, 7raleqbidv 2918 . . . . . 6
98rspccv 3051 . . . . 5
102, 9syl 16 . . . 4
11 elin 3532 . . . . . 6
12 elpwg 3808 . . . . . . 7
1312pm5.32i 620 . . . . . 6
1411, 13bitri 242 . . . . 5
15 sseq1 3371 . . . . . . . 8
1615anbi1d 687 . . . . . . 7
1716rexbidv 2728 . . . . . 6
1817rspccv 3051 . . . . 5
1914, 18syl5bir 211 . . . 4
2010, 19syl6 32 . . 3
2120exp4a 591 . 2
22213imp2 1169 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  wrex 2708   cin 3321   wss 3322  cpw 3801  cfv 5457  ctop 16963  ccld 17085  ccl 17087  cnrm 17379 This theorem is referenced by:  nrmsep2  17425  kqnrmlem1  17780  kqnrmlem2  17781  nrmr0reg  17786  nrmhmph  17831 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-nrm 17386
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