Theorem leat

Description: A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013)

	Ref	Expression
Hypotheses	leatom.b	|- B = ( Base ` K )
	leatom.l	|- .<_ = ( le ` K )
	leatom.z	|- .0. = ( 0. ` K )
	leatom.a	|- A = ( Atoms ` K )
Assertion	leat	|- ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ X .<_ P ) -> ( X = P \/ X = .0. ) )

Proof

Step	Hyp	Ref	Expression
1		leatom.b	|- B = ( Base ` K )
2		leatom.l	|- .<_ = ( le ` K )
3		leatom.z	|- .0. = ( 0. ` K )
4		leatom.a	|- A = ( Atoms ` K )
5	1 2 3 4	leatb	|- ( ( K e. OP /\ X e. B /\ P e. A ) -> ( X .<_ P <-> ( X = P \/ X = .0. ) ) )
6	5	biimpa	|- ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ X .<_ P ) -> ( X = P \/ X = .0. ) )