Theorem sncld 22522

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 ∧ 𝑃 ∈ 𝑋) → {𝑃} ∈ (Clsd‘𝐽))

Proof of Theorem sncld

Step	Hyp	Ref	Expression
1		haust1 22503	. 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Fre)
2		t1sep.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	2	t1sncld 22477	. 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑃 ∈ 𝑋) → {𝑃} ∈ (Clsd‘𝐽))
4	1, 3	sylan 580	1 ⊢ ((𝐽 ∈ Haus ∧ 𝑃 ∈ 𝑋) → {𝑃} ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {csn 4561  ∪ cuni 4839  ‘cfv 6433  Clsdccld 22167  Frect1 22458  Hauscha 22459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-topgen 17154  df-top 22043  df-topon 22060  df-cld 22170  df-t1 22465  df-haus 22466

This theorem is referenced by:  tgphaus  23268  csscld  24413  clsocv  24414  dvrec  25119  dvexp3  25142  abelth  25600  dvtanlem  35826  sncldre  42590  dirkercncflem2  43645