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Theorem t1sncld 17054
Description: In a T1 space, one-point sets are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
ist0.1  |-  X  = 
U. J
Assertion
Ref Expression
t1sncld  |-  ( ( J  e.  Fre  /\  A  e.  X )  ->  { A }  e.  ( Clsd `  J )
)

Proof of Theorem t1sncld
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ist0.1 . . . . 5  |-  X  = 
U. J
21ist1 17049 . . . 4  |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. x  e.  X  { x }  e.  ( Clsd `  J ) ) )
32simprbi 450 . . 3  |-  ( J  e.  Fre  ->  A. x  e.  X  { x }  e.  ( Clsd `  J ) )
4 sneq 3651 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
54eleq1d 2349 . . . 4  |-  ( x  =  A  ->  ( { x }  e.  ( Clsd `  J )  <->  { A }  e.  (
Clsd `  J )
) )
65rspccv 2881 . . 3  |-  ( A. x  e.  X  {
x }  e.  (
Clsd `  J )  ->  ( A  e.  X  ->  { A }  e.  ( Clsd `  J )
) )
73, 6syl 15 . 2  |-  ( J  e.  Fre  ->  ( A  e.  X  ->  { A }  e.  (
Clsd `  J )
) )
87imp 418 1  |-  ( ( J  e.  Fre  /\  A  e.  X )  ->  { A }  e.  ( Clsd `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {csn 3640   U.cuni 3827   ` cfv 5255   Topctop 16631   Clsdccld 16753   Frect1 17035
This theorem is referenced by:  cnt1  17078  lpcls  17092  sncld  17099  dnsconst  17106  t1conperf  17162  r0cld  17429  tgpt1  17800
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-3an 936  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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-t1 17042
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