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Theorem 2ndctop 17512
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 17511 . 2  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
2 simprr 735 . . . 4  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  -> 
( topGen `  x )  =  J )
3 tgcl 17036 . . . . 5  |-  ( x  e.  TopBases  ->  ( topGen `  x
)  e.  Top )
43adantr 453 . . . 4  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  -> 
( topGen `  x )  e.  Top )
52, 4eqeltrrd 2513 . . 3  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  ->  J  e.  Top )
65rexlimiva 2827 . 2  |-  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  J  e.  Top )
71, 6sylbi 189 1  |-  ( J  e.  2ndc  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214   omcom 4847   ` cfv 5456    ~<_ cdom 7109   topGenctg 13667   Topctop 16960   TopBasesctb 16964   2ndcc2ndc 17503
This theorem is referenced by:  2ndc1stc  17516  2ndcctbss  17520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-topgen 13669  df-top 16965  df-bases 16967  df-2ndc 17505
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