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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 )