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Theorem is2ndc 17509
 Description: The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
is2ndc
Distinct variable group:   ,

Proof of Theorem is2ndc
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-2ndc 17503 . . 3
21eleq2i 2500 . 2
3 simpr 448 . . . . 5
4 fvex 5742 . . . . 5
53, 4syl6eqelr 2525 . . . 4
65rexlimivw 2826 . . 3
7 eqeq2 2445 . . . . 5
87anbi2d 685 . . . 4
98rexbidv 2726 . . 3
106, 9elab3 3089 . 2
112, 10bitri 241 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  cab 2422  wrex 2706  cvv 2956   class class class wbr 4212  com 4845  cfv 5454   cdom 7107  ctg 13665  ctb 16962  c2ndc 17501 This theorem is referenced by:  2ndctop  17510  2ndci  17511  2ndcsb  17512  2ndcredom  17513  2ndc1stc  17514  2ndcrest  17517  2ndcctbss  17518  2ndcdisj  17519  2ndcomap  17521  2ndcsep  17522  dis2ndc  17523  tx2ndc  17683 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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820  df-pr 3821  df-uni 4016  df-iota 5418  df-fv 5462  df-2ndc 17503
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