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Theorem 1stctop 19738
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 2467 . . 3  |-  U. J  =  U. J
21is1stc 19736 . 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 460 1  |-  ( J  e.  1stc  ->  J  e. 
Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   A.wral 2814   E.wrex 2815    i^i cin 3475   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447   omcom 6684    ~<_ cdom 7514   Topctop 19189   1stcc1stc 19732
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-in 3483  df-ss 3490  df-pw 4012  df-uni 4246  df-1stc 19734
This theorem is referenced by:  1stcfb  19740  1stcrest  19748  1stcelcls  19756  lly1stc  19791  1stckgen  19818  tx1stc  19914
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