MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  t1sncld Unicode version

Theorem t1sncld 17344
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 17339 . . . 4  |-  ( J  e.  Fre  <->  ( J  e.  Top  /\  A. x  e.  X  { x }  e.  ( Clsd `  J ) ) )
32simprbi 451 . . 3  |-  ( J  e.  Fre  ->  A. x  e.  X  { x }  e.  ( Clsd `  J ) )
4 sneq 3785 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
54eleq1d 2470 . . . 4  |-  ( x  =  A  ->  ( { x }  e.  ( Clsd `  J )  <->  { A }  e.  (
Clsd `  J )
) )
65rspccv 3009 . . 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 419 1  |-  ( ( J  e.  Fre  /\  A  e.  X )  ->  { A }  e.  ( Clsd `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   {csn 3774   U.cuni 3975   ` cfv 5413   Topctop 16913   Clsdccld 17035   Frect1 17325
This theorem is referenced by:  cnt1  17368  lpcls  17382  sncld  17389  dnsconst  17396  t1conperf  17452  r0cld  17723  tgpt1  18100  sibfof  24607
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-t1 17332
  Copyright terms: Public domain W3C validator