Theorem cvrle

Description: The covers relation implies the "less than or equal to" relation. (Contributed by NM, 12-Oct-2011)

	Ref	Expression
Hypotheses	cvrle.b	|- B = ( Base ` K )
	cvrle.l	|- .<_ = ( le ` K )
	cvrle.c	|- C = (
Assertion	cvrle	|- ( ( ( K e. A /\ X e. B /\ Y e. B ) /\ X C Y ) -> X .<_ Y )

Proof

Step	Hyp	Ref	Expression
1		cvrle.b	|- B = ( Base ` K )
2		cvrle.l	|- .<_ = ( le ` K )
3		cvrle.c	|- C = (
4		eqid	|- ( lt ` K ) = ( lt ` K )
5	1 4 3	cvrlt	|- ( ( ( K e. A /\ X e. B /\ Y e. B ) /\ X C Y ) -> X ( lt ` K ) Y )
6	2 4	pltval	|- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X ( lt ` K ) Y <-> ( X .<_ Y /\ X =/= Y ) ) )
7	6	simprbda	|- ( ( ( K e. A /\ X e. B /\ Y e. B ) /\ X ( lt ` K ) Y ) -> X .<_ Y )
8	5 7	syldan	|- ( ( ( K e. A /\ X e. B /\ Y e. B ) /\ X C Y ) -> X .<_ Y )