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Metamath Proof Explorer

Theorem cvrntr

Description: The covers relation is not transitive. ( cvntr analog.) (Contributed by NM, 18-Jun-2012)

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

Proof

Step Hyp Ref Expression
1 cvrntr.b 
 |-  B = ( Base ` K )
2 cvrntr.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 4 ex 
 |-  ( ( K e. A /\ X e. B /\ Y e. B ) -> ( X C Y -> X ( lt ` K ) Y ) )
6 5 3adant3r3 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Y -> X ( lt ` K ) Y ) )
7 1 3 2 ltcvrntr 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X ( lt ` K ) Y /\ Y C Z ) -> -. X C Z ) )
8 6 7 syland 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X C Y /\ Y C Z ) -> -. X C Z ) )