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 )