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Theorem 2ndctop 17173
 Description: A second-countable topology is a topology. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
2ndctop

Proof of Theorem 2ndctop
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 is2ndc 17172 . 2
2 simprr 733 . . . 4
3 tgcl 16707 . . . . 5
43adantr 451 . . . 4
52, 4eqeltrrd 2358 . . 3
65rexlimiva 2662 . 2
71, 6sylbi 187 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684  wrex 2544   class class class wbr 4023  com 4656  cfv 5255   cdom 6861  ctg 13342  ctop 16631  ctb 16635  c2ndc 17164 This theorem is referenced by:  2ndc1stc  17177  2ndcctbss  17181 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-topgen 13344  df-top 16636  df-bases 16638  df-2ndc 17166
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