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Theorem regsep 20128
 Description: In a regular space, every neighborhood of a point contains a closed subneighborhood. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
regsep
Distinct variable groups:   ,   ,   ,

Proof of Theorem regsep
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isreg 20126 . . . . 5
21simprbi 462 . . . 4
3 sseq2 3464 . . . . . . . 8
43anbi2d 702 . . . . . . 7
54rexbidv 2918 . . . . . 6
65raleqbi1dv 3012 . . . . 5
76rspccv 3157 . . . 4
82, 7syl 17 . . 3
9 eleq1 2474 . . . . . 6
109anbi1d 703 . . . . 5
1110rexbidv 2918 . . . 4
1211rspccv 3157 . . 3
138, 12syl6 31 . 2
14133imp 1191 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   w3a 974   wceq 1405   wcel 1842  wral 2754  wrex 2755   wss 3414  cfv 5569  ctop 19686  ccl 19811  creg 20103 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-reg 20110 This theorem is referenced by:  regsep2  20170  regr1lem  20532  kqreglem1  20534  kqreglem2  20535  reghmph  20586  cnextcn  20859
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