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

Theorem sncld 23197
Description: A singleton is closed in a Hausdorff space. (Contributed by NM, 5-Mar-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
Hypothesis
Ref Expression
t1sep.1 𝑋 = 𝐽
Assertion
Ref Expression
sncld ((𝐽 ∈ Haus ∧ 𝑃𝑋) → {𝑃} ∈ (Clsd‘𝐽))

Proof of Theorem sncld
StepHypRef Expression
1 haust1 23178 . 2 (𝐽 ∈ Haus → 𝐽 ∈ Fre)
2 t1sep.1 . . 3 𝑋 = 𝐽
32t1sncld 23152 . 2 ((𝐽 ∈ Fre ∧ 𝑃𝑋) → {𝑃} ∈ (Clsd‘𝐽))
41, 3sylan 579 1 ((𝐽 ∈ Haus ∧ 𝑃𝑋) → {𝑃} ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {csn 4620   cuni 4899  cfv 6533  Clsdccld 22842  Frect1 23133  Hauscha 23134
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541  df-topgen 17388  df-top 22718  df-topon 22735  df-cld 22845  df-t1 23140  df-haus 23141
This theorem is referenced by:  tgphaus  23943  csscld  25099  clsocv  25100  dvrec  25809  dvexp3  25832  abelth  26295  dvtanlem  37027  sncldre  44217  dirkercncflem2  45305
  Copyright terms: Public domain W3C validator