Metamath Proof Explorer


Theorem cvrne

Description: The covers relation implies inequality. (Contributed by NM, 13-Oct-2011)

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

Proof

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