Metamath Proof Explorer

Theorem ltltncvr

Description: A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012)

Ref Expression
Hypotheses ltltncvr.b 
|- B = ( Base ` K )
ltltncvr.s 
|- .< = ( lt ` K )
ltltncvr.c 
|- C = (
Assertion ltltncvr 
|- ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .< 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 simpll 
 |-  ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> K e. A )
5 simplr1 
 |-  ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> X e. B )
6 simplr3 
 |-  ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> Z e. B )
7 simplr2 
 |-  ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> Y e. B )
8 simpr 
 |-  ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> X C Z )
9 1 2 3 cvrnbtwn 
 |-  ( ( K e. A /\ ( X e. B /\ Z e. B /\ Y e. B ) /\ X C Z ) -> -. ( X .< Y /\ Y .< Z ) )
10 4 5 6 7 8 9 syl131anc 
 |-  ( ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) /\ X C Z ) -> -. ( X .< Y /\ Y .< Z ) )
11 10 ex 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Z -> -. ( X .< Y /\ Y .< Z ) ) )
12 11 con2d 
 |-  ( ( K e. A /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .< Y /\ Y .< Z ) -> -. X C Z ) )