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Theorem 1stctop 19046
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 19044 . 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 1756   A.wral 2714   E.wrex 2715    i^i cin 3326   ~Pcpw 3859   U.cuni 4090   class class class wbr 4291   omcom 6475    ~<_ cdom 7307   Topctop 18497   1stcc1stc 19040
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-in 3334  df-ss 3341  df-pw 3861  df-uni 4091  df-1stc 19042
This theorem is referenced by:  1stcfb  19048  1stcrest  19056  1stcelcls  19064  lly1stc  19099  1stckgen  19126  tx1stc  19222
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