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Theorem sncld 22874
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 22855 . 2 (𝐽 ∈ Haus → 𝐽 ∈ Fre)
2 t1sep.1 . . 3 𝑋 = 𝐽
32t1sncld 22829 . 2 ((𝐽 ∈ Fre ∧ 𝑃𝑋) → {𝑃} ∈ (Clsd‘𝐽))
41, 3sylan 580 1 ((𝐽 ∈ Haus ∧ 𝑃𝑋) → {𝑃} ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {csn 4628   cuni 4908  cfv 6543  Clsdccld 22519  Frect1 22810  Hauscha 22811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-topgen 17388  df-top 22395  df-topon 22412  df-cld 22522  df-t1 22817  df-haus 22818
This theorem is referenced by:  tgphaus  23620  csscld  24765  clsocv  24766  dvrec  25471  dvexp3  25494  abelth  25952  dvtanlem  36532  sncldre  43719  dirkercncflem2  44810
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