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Theorem t1top 17316
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 2387 . . 3  |-  U. J  =  U. J
21ist1 17307 . 2  |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. x  e.  U. J { x }  e.  ( Clsd `  J ) ) )
32simplbi 447 1  |-  ( J  e.  Fre  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   A.wral 2649   {csn 3757   U.cuni 3957   ` cfv 5394   Topctop 16881   Clsdccld 17003   Frect1 17293
This theorem is referenced by:  t1t0  17334  lpcls  17350  perfcls  17351  restt1  17353  t1sep2  17355  sst1  17360  t1conperf  17420  t1hmph  17744  onint1  25913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-t1 17300
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