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Theorem nrmsep3 20448
 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 20428 . . . . . 6
21simprbi 471 . . . . 5
3 pweq 3945 . . . . . . . 8
43ineq2d 3625 . . . . . . 7
5 sseq2 3440 . . . . . . . . 9
65anbi2d 718 . . . . . . . 8
76rexbidv 2892 . . . . . . 7
84, 7raleqbidv 2987 . . . . . 6
98rspccv 3133 . . . . 5
102, 9syl 17 . . . 4
11 elin 3608 . . . . . 6
12 elpwg 3950 . . . . . . 7
1312pm5.32i 649 . . . . . 6
1411, 13bitri 257 . . . . 5
15 sseq1 3439 . . . . . . . 8
1615anbi1d 719 . . . . . . 7
1716rexbidv 2892 . . . . . 6
1817rspccv 3133 . . . . 5
1914, 18syl5bir 226 . . . 4
2010, 19syl6 33 . . 3
2120exp4a 617 . 2
22213imp2 1248 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   w3a 1007   wceq 1452   wcel 1904  wral 2756  wrex 2757   cin 3389   wss 3390  cpw 3942  cfv 5589  ctop 19994  ccld 20108  ccl 20110  cnrm 20403 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-nrm 20410 This theorem is referenced by:  nrmsep2  20449  kqnrmlem1  20835  kqnrmlem2  20836  nrmr0reg  20841  nrmhmph  20886
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