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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  |-  ( J  e.  2ndc  ->  J  e. 
Top )

Proof of Theorem 2ndctop
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 is2ndc 17172 . 2  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
2 simprr 733 . . . 4  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  -> 
( topGen `  x )  =  J )
3 tgcl 16707 . . . . 5  |-  ( x  e.  TopBases  ->  ( topGen `  x
)  e.  Top )
43adantr 451 . . . 4  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  -> 
( topGen `  x )  e.  Top )
52, 4eqeltrrd 2358 . . 3  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  ->  J  e.  Top )
65rexlimiva 2662 . 2  |-  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  J  e.  Top )
71, 6sylbi 187 1  |-  ( J  e.  2ndc  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   omcom 4656   ` cfv 5255    ~<_ cdom 6861   topGenctg 13342   Topctop 16631   TopBasesctb 16635   2ndcc2ndc 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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