Theorem atlt

Description: Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012)

	Ref	Expression
Hypotheses	atlt.s	|- .< = ( lt ` K )
	atlt.j	|- .\/ = ( join ` K )
	atlt.a	|- A = ( Atoms ` K )
Assertion	atlt	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .< ( P .\/ Q ) <-> P =/= Q ) )

Proof

Step	Hyp	Ref	Expression
1		atlt.s	|- .< = ( lt ` K )
2		atlt.j	|- .\/ = ( join ` K )
3		atlt.a	|- A = ( Atoms ` K )
4		simp1	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> K e. HL )
5		simp2	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> P e. A )
6		simp3	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> Q e. A )
7		eqid	|- (
8	1 2 3 7	atltcvr	|- ( ( K e. HL /\ ( P e. A /\ P e. A /\ Q e. A ) ) -> ( P .< ( P .\/ Q ) <-> P (
9	4 5 5 6 8	syl13anc	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .< ( P .\/ Q ) <-> P (
10	2 7 3	atcvr1	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P (
11	9 10	bitr4d	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .< ( P .\/ Q ) <-> P =/= Q ) )