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Theorem 1stctop 17537
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 2442 . . 3  |-  U. J  =  U. J
21is1stc 17535 . 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 448 1  |-  ( J  e.  1stc  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727   A.wral 2711   E.wrex 2712    i^i cin 3305   ~Pcpw 3823   U.cuni 4039   class class class wbr 4237   omcom 4874    ~<_ cdom 7136   Topctop 16989   1stcc1stc 17531
This theorem is referenced by:  1stcfb  17539  1stcrest  17547  1stcelcls  17555  lly1stc  17590  1stckgen  17617  tx1stc  17713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-in 3313  df-ss 3320  df-pw 3825  df-uni 4040  df-1stc 17533
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