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Theorem 1stctop 17275
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 2358 . . 3  |-  U. J  =  U. J
21is1stc 17273 . 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 1710   A.wral 2619   E.wrex 2620    i^i cin 3227   ~Pcpw 3701   U.cuni 3908   class class class wbr 4104   omcom 4738    ~<_ cdom 6949   Topctop 16737   1stcc1stc 17269
This theorem is referenced by:  1stcfb  17277  1stcrest  17285  1stcelcls  17293  lly1stc  17328  1stckgen  17355  tx1stc  17450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-in 3235  df-ss 3242  df-pw 3703  df-uni 3909  df-1stc 17271
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