Theorem clsss3

Description: The closure of a subset of a topological space is included in the space. (Contributed by NM, 26-Feb-2007)

	Ref	Expression
Hypothesis	clscld.1	|- X = U. J
Assertion	clsss3	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ X )

Proof

Step	Hyp	Ref	Expression
1		clscld.1	|- X = U. J
2	1	clscld	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) e. ( Clsd ` J ) )
3	1	cldss	|- ( ( ( cls ` J ) ` S ) e. ( Clsd ` J ) -> ( ( cls ` J ) ` S ) C_ X )
4	2 3	syl	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ X )