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

Theorem t1sncld 23250
Description: In a T1 space, singletons are closed. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
ist0.1 𝑋 = 𝐽
Assertion
Ref Expression
t1sncld ((𝐽 ∈ Fre ∧ 𝐴𝑋) → {𝐴} ∈ (Clsd‘𝐽))

Proof of Theorem t1sncld
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ist0.1 . . . 4 𝑋 = 𝐽
21ist1 23245 . . 3 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽)))
3 sneq 4642 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
43eleq1d 2814 . . . 4 (𝑥 = 𝐴 → ({𝑥} ∈ (Clsd‘𝐽) ↔ {𝐴} ∈ (Clsd‘𝐽)))
54rspccv 3608 . . 3 (∀𝑥𝑋 {𝑥} ∈ (Clsd‘𝐽) → (𝐴𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
62, 5simplbiim 503 . 2 (𝐽 ∈ Fre → (𝐴𝑋 → {𝐴} ∈ (Clsd‘𝐽)))
76imp 405 1 ((𝐽 ∈ Fre ∧ 𝐴𝑋) → {𝐴} ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3058  {csn 4632   cuni 4912  cfv 6553  Topctop 22815  Clsdccld 22940  Frect1 23231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-t1 23238
This theorem is referenced by:  cnt1  23274  lpcls  23288  sncld  23295  dnsconst  23302  t1connperf  23360  r0cld  23662  tgpt1  24042  sibfinima  33992  sibfof  33993
  Copyright terms: Public domain W3C validator