Theorem lubub

Description: The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011)

	Ref	Expression
Hypotheses	lublem.b	|- B = ( Base ` K )
	lublem.l	|- .<_ = ( le ` K )
	lublem.u	|- U = ( lub ` K )
Assertion	lubub	|- ( ( K e. CLat /\ S C_ B /\ X e. S ) -> X .<_ ( U ` S ) )

Proof

Step	Hyp	Ref	Expression
1		lublem.b	|- B = ( Base ` K )
2		lublem.l	|- .<_ = ( le ` K )
3		lublem.u	|- U = ( lub ` K )
4	1 2 3	lublem	|- ( ( K e. CLat /\ S C_ B ) -> ( A. y e. S y .<_ ( U ` S ) /\ A. z e. B ( A. y e. S y .<_ z -> ( U ` S ) .<_ z ) ) )
5	4	simpld	|- ( ( K e. CLat /\ S C_ B ) -> A. y e. S y .<_ ( U ` S ) )
6		breq1	|- ( y = X -> ( y .<_ ( U ` S ) <-> X .<_ ( U ` S ) ) )
7	6	rspccva	|- ( ( A. y e. S y .<_ ( U ` S ) /\ X e. S ) -> X .<_ ( U ` S ) )
8	5 7	stoic3	|- ( ( K e. CLat /\ S C_ B /\ X e. S ) -> X .<_ ( U ` S ) )