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Theorem sncld 23395
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 23376 . 2 (𝐽 ∈ Haus → 𝐽 ∈ Fre)
2 t1sep.1 . . 3 𝑋 = 𝐽
32t1sncld 23350 . 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 395   = wceq 1537  wcel 2106  {csn 4631   cuni 4912  cfv 6563  Clsdccld 23040  Frect1 23331  Hauscha 23332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-topgen 17490  df-top 22916  df-topon 22933  df-cld 23043  df-t1 23338  df-haus 23339
This theorem is referenced by:  tgphaus  24141  csscld  25297  clsocv  25298  dvrec  26008  dvexp3  26031  abelth  26500  dvtanlem  37656  readvrec2  42370  sncldre  44982  dirkercncflem2  46060
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