MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2ndctop Structured version   Unicode version

Theorem 2ndctop 19051
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 19050 . 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 18574 . . . . 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 2518 . . 3  |-  ( ( x  e.  TopBases  /\  (
x  ~<_  om  /\  ( topGen `
 x )  =  J ) )  ->  J  e.  Top )
65rexlimiva 2836 . 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 1369    e. wcel 1756   E.wrex 2716   class class class wbr 4292   ` cfv 5418   omcom 6476    ~<_ cdom 7308   topGenctg 14376   Topctop 18498   TopBasesctb 18502   2ndcc2ndc 19042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-topgen 14382  df-top 18503  df-bases 18505  df-2ndc 19044
This theorem is referenced by:  2ndc1stc  19055  2ndcctbss  19059
  Copyright terms: Public domain W3C validator