Theorem opncld

Description: The complement of an open set is closed. (Contributed by NM, 6-Oct-2006)

	Ref	Expression
Hypothesis	iscld.1	|- X = U. J
Assertion	opncld	|- ( ( J e. Top /\ S e. J ) -> ( X \ S ) e. ( Clsd ` J ) )

Proof

Step	Hyp	Ref	Expression
1		iscld.1	|- X = U. J
2		simpr	|- ( ( J e. Top /\ S e. J ) -> S e. J )
3	1	eltopss	|- ( ( J e. Top /\ S e. J ) -> S C_ X )
4	1	isopn2	|- ( ( J e. Top /\ S C_ X ) -> ( S e. J <-> ( X \ S ) e. ( Clsd ` J ) ) )
5	3 4	syldan	|- ( ( J e. Top /\ S e. J ) -> ( S e. J <-> ( X \ S ) e. ( Clsd ` J ) ) )
6	2 5	mpbid	|- ( ( J e. Top /\ S e. J ) -> ( X \ S ) e. ( Clsd ` J ) )