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Metamath Proof Explorer

Theorem pltnle

Description: "Less than" implies not converse "less than or equal to". (Contributed by NM, 18-Oct-2011)

Ref Expression
Hypotheses pleval2.b 
|- B = ( Base ` K )
pleval2.l 
|- .<_ = ( le ` K )
pleval2.s 
|- .< = ( lt ` K )
Assertion pltnle 
|- ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .< Y ) -> -. Y .<_ X )

Proof

Step Hyp Ref Expression
1 pleval2.b 
 |-  B = ( Base ` K )
2 pleval2.l 
 |-  .<_ = ( le ` K )
3 pleval2.s 
 |-  .< = ( lt ` K )
4 2 3 pltval 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( X .<_ Y /\ X =/= Y ) ) )
5 1 2 posasymb 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )
6 5 biimpd 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) -> X = Y ) )
7 6 expdimp 
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .<_ Y ) -> ( Y .<_ X -> X = Y ) )
8 7 necon3ad 
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .<_ Y ) -> ( X =/= Y -> -. Y .<_ X ) )
9 8 expimpd 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ X =/= Y ) -> -. Y .<_ X ) )
10 4 9 sylbid 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y -> -. Y .<_ X ) )
11 10 imp 
 |-  ( ( ( K e. Poset /\ X e. B /\ Y e. B ) /\ X .< Y ) -> -. Y .<_ X )