Metamath Proof Explorer


Theorem cvrnle

Description: The covers relation implies the negation of the converse "less than or equal to" relation. (Contributed by NM, 18-Oct-2011)

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

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. Poset /\ X e. B /\ Y e. B ) /\ X C Y ) -> X ( lt ` K ) Y )
6 1 2 4 pltnle
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X ( lt ` K ) Y ) -> -. Y .<_ X )
7 5 6 syldan
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X C Y ) -> -. Y .<_ X )