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Theorem haustop 17397
Description: A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.)
Assertion
Ref Expression
haustop  |-  ( J  e.  Haus  ->  J  e. 
Top )

Proof of Theorem haustop
Dummy variables  x  y  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3  |-  U. J  =  U. J
21ishaus 17388 . 2  |-  ( J  e.  Haus  <->  ( J  e. 
Top  /\  A. x  e.  U. J A. y  e.  U. J ( x  =/=  y  ->  E. n  e.  J  E. m  e.  J  ( x  e.  n  /\  y  e.  m  /\  (
n  i^i  m )  =  (/) ) ) ) )
32simplbi 448 1  |-  ( J  e.  Haus  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    i^i cin 3321   (/)c0 3630   U.cuni 4017   Topctop 16960   Hauscha 17374
This theorem is referenced by:  haust1  17418  resthaus  17434  sshaus  17441  lmmo  17446  hauscmplem  17471  hauscmp  17472  hauslly  17557  hausllycmp  17559  kgenhaus  17578  pthaus  17672  txhaus  17681  xkohaus  17687  haushmph  17826  cmphaushmeo  17834  hausflim  18015  hauspwpwf1  18021  hauspwpwdom  18022  hausflf  18031  cnextfun  18097  cnextfvval  18098  cnextf  18099  cnextcn  18100  cnextfres  18101  hausgraph  27510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
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 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-uni 4018  df-haus 17381
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