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 ) )