Theorem atncmp

Description: Frequently-used variation of atcmp . (Contributed by NM, 29-Jun-2012)

	Ref	Expression
Hypotheses	atcmp.l	|- .<_ = ( le ` K )
	atcmp.a	|- A = ( Atoms ` K )
Assertion	atncmp	|- ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( -. P .<_ Q <-> P =/= Q ) )

Proof

Step	Hyp	Ref	Expression
1		atcmp.l	|- .<_ = ( le ` K )
2		atcmp.a	|- A = ( Atoms ` K )
3	1 2	atcmp	|- ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( P .<_ Q <-> P = Q ) )
4	3	necon3bbid	|- ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( -. P .<_ Q <-> P =/= Q ) )