Theorem isat2

Description: The predicate "is an atom". ( elatcv0 analog.) (Contributed by NM, 18-Jun-2012)

	Ref	Expression
Hypotheses	isatom.b	|- B = ( Base ` K )
	isatom.z	|- .0. = ( 0. ` K )
	isatom.c	|- C = (
	isatom.a	|- A = ( Atoms ` K )
Assertion	isat2	|- ( ( K e. D /\ P e. B ) -> ( P e. A <-> .0. C P ) )

Proof

Step	Hyp	Ref	Expression
1		isatom.b	|- B = ( Base ` K )
2		isatom.z	|- .0. = ( 0. ` K )
3		isatom.c	|- C = (
4		isatom.a	|- A = ( Atoms ` K )
5	1 2 3 4	isat	|- ( K e. D -> ( P e. A <-> ( P e. B /\ .0. C P ) ) )
6	5	baibd	|- ( ( K e. D /\ P e. B ) -> ( P e. A <-> .0. C P ) )