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Theorem haustop 17059
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 2283 . . 3  |-  U. J  =  U. J
21ishaus 17050 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151   (/)c0 3455   U.cuni 3827   Topctop 16631   Hauscha 17036
This theorem is referenced by:  haust1  17080  resthaus  17096  sshaus  17103  lmmo  17108  hauscmplem  17133  hauscmp  17134  hauslly  17218  hausllycmp  17220  kgenhaus  17239  pthaus  17332  txhaus  17341  xkohaus  17347  haushmph  17483  cmphaushmeo  17491  hausflim  17676  hauspwpwf1  17682  hauspwpwdom  17683  hausflf  17692  flfneih  25560  limfn3  25567  cmptdst2  25571  hausgraph  27531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-uni 3828  df-haus 17043
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