Theorem cvrlt

Description: The covers relation implies the less-than relation. ( cvpss analog.) (Contributed by NM, 8-Oct-2011)

	Ref	Expression
Hypotheses	cvrfval.b	|- B = ( Base ` K )
	cvrfval.s	|- .< = ( lt ` K )
	cvrfval.c	|- C = (
Assertion	cvrlt	|- ( ( ( K e. A /\ X e. B /\ Y e. B ) /\ X C Y ) -> X .< Y )

Proof

Step	Hyp	Ref	Expression
1		cvrfval.b	|- B = ( Base ` K )
2		cvrfval.s	|- .< = ( lt ` K )
3		cvrfval.c	|- C = (
4	1 2 3	cvrval	|- ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X C Y <-> ( X .< Y /\ -. E. z e. B ( X .< z /\ z .< Y ) ) ) )
5	4	simprbda	|- ( ( ( K e. A /\ X e. B /\ Y e. B ) /\ X C Y ) -> X .< Y )