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Theorem t1top 19998
Description: A T1 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.)
Assertion
Ref Expression
t1top  |-  ( J  e.  Fre  ->  J  e.  Top )

Proof of Theorem t1top
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2454 . . 3  |-  U. J  =  U. J
21ist1 19989 . 2  |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. x  e.  U. J { x }  e.  ( Clsd `  J ) ) )
32simplbi 458 1  |-  ( J  e.  Fre  ->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823   A.wral 2804   {csn 4016   U.cuni 4235   ` cfv 5570   Topctop 19561   Clsdccld 19684   Frect1 19975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-t1 19982
This theorem is referenced by:  t1t0  20016  lpcls  20032  perfcls  20033  restt1  20035  t1sep2  20037  sst1  20042  t1conperf  20103  t1hmph  20458  qtopt1  28073  onint1  30142
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