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Metamath Proof Explorer

Theorem pltn2lp

Description: The less-than relation has no 2-cycle loops. ( pssn2lp analog.) (Contributed by NM, 2-Dec-2011)

Ref Expression
Hypotheses pltnlt.b 
|- B = ( Base ` K )
pltnlt.s 
|- .< = ( lt ` K )
Assertion pltn2lp 
|- ( ( 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 4 ex 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y -> -. Y ( le ` K ) X ) )
6 3 2 pltle 
 |-  ( ( K e. Poset /\ Y e. B /\ X e. B ) -> ( Y .< X -> Y ( le ` K ) X ) )
7 6 3com23 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( Y .< X -> Y ( le ` K ) X ) )
8 5 7 nsyld 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y -> -. Y .< X ) )
9 imnan 
 |-  ( ( X .< Y -> -. Y .< X ) <-> -. ( X .< Y /\ Y .< X ) )
10 8 9 sylib 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> -. ( X .< Y /\ Y .< X ) )