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Theorem t1top 17074
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 2296 . . 3  |-  U. J  =  U. J
21ist1 17065 . 2  |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. x  e.  U. J { x }  e.  ( Clsd `  J ) ) )
32simplbi 446 1  |-  ( J  e.  Fre  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   A.wral 2556   {csn 3653   U.cuni 3843   ` cfv 5271   Topctop 16647   Clsdccld 16769   Frect1 17051
This theorem is referenced by:  t1t0  17092  lpcls  17108  perfcls  17109  restt1  17111  t1sep2  17113  sst1  17118  t1conperf  17178  t1hmph  17498  onint1  24960
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-3an 936  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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-t1 17058
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