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Theorem t1sncld 17395
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 17390 . . . 4  |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. x  e.  X  { x }  e.  ( Clsd `  J ) ) )
32simprbi 452 . . 3  |-  ( J  e.  Fre  ->  A. x  e.  X  { x }  e.  ( Clsd `  J ) )
4 sneq 3827 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
54eleq1d 2504 . . . 4  |-  ( x  =  A  ->  ( { x }  e.  ( Clsd `  J )  <->  { A }  e.  (
Clsd `  J )
) )
65rspccv 3051 . . 3  |-  ( A. x  e.  X  {
x }  e.  (
Clsd `  J )  ->  ( A  e.  X  ->  { A }  e.  ( Clsd `  J )
) )
73, 6syl 16 . 2  |-  ( J  e.  Fre  ->  ( A  e.  X  ->  { A }  e.  (
Clsd `  J )
) )
87imp 420 1  |-  ( ( J  e.  Fre  /\  A  e.  X )  ->  { A }  e.  ( Clsd `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {csn 3816   U.cuni 4017   ` cfv 5457   Topctop 16963   Clsdccld 17085   Frect1 17376
This theorem is referenced by:  cnt1  17419  lpcls  17433  sncld  17440  dnsconst  17447  t1conperf  17504  r0cld  17775  tgpt1  18152  sibfof  24659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-t1 17383
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