Metamath Proof Explorer


Theorem pltnlt

Description: The less-than relation implies the negation of its inverse. (Contributed by NM, 18-Oct-2011)

Ref Expression
Hypotheses pltnlt.b
|- B = ( Base ` K )
pltnlt.s
|- .< = ( lt ` K )
Assertion pltnlt
|- ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .< Y ) -> -. Y .< X )

Proof

Step Hyp Ref Expression
1 pltnlt.b
 |-  B = ( Base ` K )
2 pltnlt.s
 |-  .< = ( lt ` K )
3 eqid
 |-  ( le ` K ) = ( le ` K )
4 1 3 2 pltnle
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .< Y ) -> -. Y ( le ` K ) X )
5 3 2 pltle
 |-  ( ( K e. Poset /\ Y e. B /\ X e. B ) -> ( Y .< X -> Y ( le ` K ) X ) )
6 5 3com23
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( Y .< X -> Y ( le ` K ) X ) )
7 6 adantr
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .< Y ) -> ( Y .< X -> Y ( le ` K ) X ) )
8 4 7 mtod
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .< Y ) -> -. Y .< X )