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Theorem 2ndctop 20033
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 20032 . 2  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
2 simprr 755 . . . 4  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  -> 
( topGen `  x )  =  J )
3 tgcl 19556 . . . . 5  |-  ( x  e.  TopBases  ->  ( topGen `  x
)  e.  Top )
43adantr 463 . . . 4  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  -> 
( topGen `  x )  e.  Top )
52, 4eqeltrrd 2471 . . 3  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  ->  J  e.  Top )
65rexlimiva 2870 . 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 367    = wceq 1399    e. wcel 1826   E.wrex 2733   class class class wbr 4367   ` cfv 5496   omcom 6599    ~<_ cdom 7433   topGenctg 14845   Topctop 19479   TopBasesctb 19483   2ndcc2ndc 20024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-topgen 14851  df-top 19484  df-bases 19486  df-2ndc 20026
This theorem is referenced by:  2ndc1stc  20037  2ndcctbss  20041
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