Theorem t1sncld 21130

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.

Step	Hyp	Ref	Expression
1		ist0.1	. . . 4 |- X = U. J
2	1	ist1 21125	. . 3 |- ( J e. Fre <-> ( J e. Top /\ A. x e. X { x } e. ( Clsd ` J ) ) )
3		sneq 4187	. . . . 5 |- ( x = A -> { x } = { A } )
4	3	eleq1d 2686	. . . 4 |- ( x = A -> ( { x } e. ( Clsd ` J ) <-> { A } e. ( Clsd ` J ) ) )
5	4	rspccv 3306	. . 3 |- ( A. x e. X { x } e. ( Clsd ` J ) -> ( A e. X -> { A } e. ( Clsd ` J ) ) )
6	2, 5	simplbiim 659	. 2 |- ( J e. Fre -> ( A e. X -> { A } e. ( Clsd ` J ) ) )
7	6	imp 445	1 |- ( ( J e. Fre /\ A e. X ) -> { A } e. ( Clsd ` J ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {csn 4177   U.cuni 4436   ` cfv 5888   Topctop 20698   Clsdccld 20820   Frect1 21111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-t1 21118

This theorem is referenced by:  cnt1  21154  lpcls  21168  sncld  21175  dnsconst  21182  t1connperf  21239  r0cld  21541  tgpt1  21921  sibfinima  30401  sibfof  30402