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Theorem 1stctop 17185
Description: A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
Assertion
Ref Expression
1stctop  |-  ( J  e.  1stc  ->  J  e. 
Top )

Proof of Theorem 1stctop
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . 3  |-  U. J  =  U. J
21is1stc 17183 . 2  |-  ( J  e.  1stc  <->  ( J  e. 
Top  /\  A. x  e.  U. J E. y  e.  ~P  J ( y  ~<_  om  /\  A. z  e.  J  ( x  e.  z  ->  x  e. 
U. ( y  i^i 
~P z ) ) ) ) )
32simplbi 446 1  |-  ( J  e.  1stc  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039   omcom 4672    ~<_ cdom 6877   Topctop 16647   1stcc1stc 17179
This theorem is referenced by:  1stcfb  17187  1stcrest  17195  1stcelcls  17203  lly1stc  17238  1stckgen  17265  tx1stc  17360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844  df-1stc 17181
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