Theorem clatlubcl

Description: Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 14-Sep-2011)

	Ref	Expression
Hypotheses	clatlubcl.b	|- B = ( Base ` K )
	clatlubcl.u	|- U = ( lub ` K )
Assertion	clatlubcl	|- ( ( K e. CLat /\ S C_ B ) -> ( U ` S ) e. B )

Proof

Step	Hyp	Ref	Expression
1		clatlubcl.b	|- B = ( Base ` K )
2		clatlubcl.u	|- U = ( lub ` K )
3		eqid	|- ( glb ` K ) = ( glb ` K )
4	1 2 3	clatlem	|- ( ( K e. CLat /\ S C_ B ) -> ( ( U ` S ) e. B /\ ( ( glb ` K ) ` S ) e. B ) )
5	4	simpld	|- ( ( K e. CLat /\ S C_ B ) -> ( U ` S ) e. B )