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Theorem t1sncld 17154
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 17149 . . . 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 3727 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
54eleq1d 2424 . . . 4  |-  ( x  =  A  ->  ( { x }  e.  ( Clsd `  J )  <->  { A }  e.  (
Clsd `  J )
) )
65rspccv 2957 . . 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 1642    e. wcel 1710   A.wral 2619   {csn 3716   U.cuni 3906   ` cfv 5334   Topctop 16731   Clsdccld 16853   Frect1 17135
This theorem is referenced by:  cnt1  17178  lpcls  17192  sncld  17199  dnsconst  17206  t1conperf  17262  r0cld  17529  tgpt1  17896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-iota 5298  df-fv 5342  df-t1 17142
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