Theorem leat3

Description: A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012)

	Ref	Expression
Hypotheses	leatom.b	|- B = ( Base ` K )
	leatom.l	|- .<_ = ( le ` K )
	leatom.z	|- .0. = ( 0. ` K )
	leatom.a	|- A = ( Atoms ` K )
Assertion	leat3	|- ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ X .<_ P ) -> ( X e. A \/ 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	leat	|- ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ X .<_ P ) -> ( X = P \/ X = .0. ) )
6		simpl3	|- ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ X .<_ P ) -> P e. A )
7		eleq1a	|- ( P e. A -> ( X = P -> X e. A ) )
8	6 7	syl	|- ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ X .<_ P ) -> ( X = P -> X e. A ) )
9	8	orim1d	|- ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ X .<_ P ) -> ( ( X = P \/ X = .0. ) -> ( X e. A \/ X = .0. ) ) )
10	5 9	mpd	|- ( ( ( K e. OP /\ X e. B /\ P e. A ) /\ X .<_ P ) -> ( X e. A \/ X = .0. ) )