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Theorem 2ndctop 19816
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 19815 . 2  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
2 simprr 756 . . . 4  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  -> 
( topGen `  x )  =  J )
3 tgcl 19339 . . . . 5  |-  ( x  e.  TopBases  ->  ( topGen `  x
)  e.  Top )
43adantr 465 . . . 4  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  -> 
( topGen `  x )  e.  Top )
52, 4eqeltrrd 2556 . . 3  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  ->  J  e.  Top )
65rexlimiva 2955 . 2  |-  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  J  e.  Top )
71, 6sylbi 195 1  |-  ( J  e.  2ndc  ->  J  e. 
Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   class class class wbr 4453   ` cfv 5594   omcom 6695    ~<_ cdom 7526   topGenctg 14710   Topctop 19263   TopBasesctb 19267   2ndcc2ndc 19807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-topgen 14716  df-top 19268  df-bases 19270  df-2ndc 19809
This theorem is referenced by:  2ndc1stc  19820  2ndcctbss  19824
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