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Theorem sncld 21223
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 21204 . 2 (𝐽 ∈ Haus → 𝐽 ∈ Fre)
2 t1sep.1 . . 3 𝑋 = 𝐽
32t1sncld 21178 . 2 ((𝐽 ∈ Fre ∧ 𝑃𝑋) → {𝑃} ∈ (Clsd‘𝐽))
41, 3sylan 487 1 ((𝐽 ∈ Haus ∧ 𝑃𝑋) → {𝑃} ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {csn 4210   cuni 4468  cfv 5926  Clsdccld 20868  Frect1 21159  Hauscha 21160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151  df-top 20747  df-topon 20764  df-cld 20871  df-t1 21166  df-haus 21167
This theorem is referenced by:  tgphaus  21967  csscld  23094  clsocv  23095  dvrec  23763  dvexp3  23786  abelth  24240  dvtanlem  33589  sncldre  39522  dirkercncflem2  40639
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