Theorem atleneN

Description: Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	atlene.l	|- .<_ = ( le ` K )
	atlene.j	|- .\/ = ( join ` K )
	atlene.a	|- A = ( Atoms ` K )
Assertion	atleneN	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> Q =/= R )

Proof

Step	Hyp	Ref	Expression
1		atlene.l	|- .<_ = ( le ` K )
2		atlene.j	|- .\/ = ( join ` K )
3		atlene.a	|- A = ( Atoms ` K )
4		eqid	|- (
5	1 2 4 3	atcvrj1	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> P (
6	2 4 3	atcvrneN	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P ( Q =/= R )
7	5 6	syld3an3	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> Q =/= R )