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Theorem haustop 17075
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 2296 . . 3  |-  U. J  =  U. J
21ishaus 17066 . 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 446 1  |-  ( J  e.  Haus  ->  J  e. 
Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    i^i cin 3164   (/)c0 3468   U.cuni 3843   Topctop 16647   Hauscha 17052
This theorem is referenced by:  haust1  17096  resthaus  17112  sshaus  17119  lmmo  17124  hauscmplem  17149  hauscmp  17150  hauslly  17234  hausllycmp  17236  kgenhaus  17255  pthaus  17348  txhaus  17357  xkohaus  17363  haushmph  17499  cmphaushmeo  17507  hausflim  17692  hauspwpwf1  17698  hauspwpwdom  17699  hausflf  17708  flfneih  25663  limfn3  25670  cmptdst2  25674  hausgraph  27634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-uni 3844  df-haus 17059
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