Metamath Proof Explorer


Theorem sncld

Description: A singleton is closed in a Hausdorff space. (Contributed by NM, 5-Mar-2007) (Revised by Mario Carneiro, 24-Aug-2015)

Ref Expression
Hypothesis t1sep.1
|- X = U. J
Assertion sncld
|- ( ( J e. Haus /\ P e. X ) -> { P } e. ( Clsd ` J ) )

Proof

Step Hyp Ref Expression
1 t1sep.1
 |-  X = U. J
2 haust1
 |-  ( J e. Haus -> J e. Fre )
3 1 t1sncld
 |-  ( ( J e. Fre /\ P e. X ) -> { P } e. ( Clsd ` J ) )
4 2 3 sylan
 |-  ( ( J e. Haus /\ P e. X ) -> { P } e. ( Clsd ` J ) )