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Metamath Proof Explorer

Theorem ltcvrntr

Description: Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012)

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

Proof

Step Hyp Ref Expression
1 ltltncvr.b 
 |-  B = ( Base ` K )
2 ltltncvr.s 
 |-  .< = ( lt ` K )
3 ltltncvr.c 
 |-  C = (
4 1 2 3 cvrlt 
 |-  ( ( ( K e. A /\ Y e. B /\ Z e. B ) /\ Y C Z ) -> Y .< Z )
5 4 ex 
 |-  ( ( K e. A /\ Y e. B /\ Z e. B ) -> ( Y C Z -> Y .< Z ) )
6 5 3adant3r1 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y C Z -> Y .< Z ) )
7 1 2 3 ltltncvr 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .< Z ) -> -. X C Z ) )
8 6 7 sylan2d 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y C Z ) -> -. X C Z ) )