Theorem latasymb

Description: A lattice ordering is asymmetric. ( eqss analog.) (Contributed by NM, 22-Oct-2011)

	Ref	Expression
Hypotheses	latref.b	|- B = ( Base ` K )
	latref.l	|- .<_ = ( le ` K )
Assertion	latasymb	|- ( ( 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		latpos	|- ( K e. Lat -> K e. Poset )
4	1 2	posasymb	|- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )
5	3 4	syl3an1	|- ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )