Theorem latasym

Description: A lattice ordering is asymmetric. ( eqss analog.) (Contributed by NM, 8-Oct-2011)

	Ref	Expression
Hypotheses	latref.b	|- B = ( Base ` K )
	latref.l	|- .<_ = ( le ` K )
Assertion	latasym	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) -> X = Y ) )

Proof

Step	Hyp	Ref	Expression
1		latref.b	|- B = ( Base ` K )
2		latref.l	|- .<_ = ( le ` K )
3	1 2	latasymb	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )
4	3	biimpd	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) -> X = Y ) )