Theorem isat

Description: The predicate "is an atom". ( ela analog.) (Contributed by NM, 18-Sep-2011)

	Ref	Expression
Hypotheses	isatom.b	|- B = ( Base ` K )
	isatom.z	|- .0. = ( 0. ` K )
	isatom.c	|- C = (
	isatom.a	|- A = ( Atoms ` K )
Assertion	isat	|- ( K e. D -> ( P e. A <-> ( P e. B /\ .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	pats	|- ( K e. D -> A = { x e. B | .0. C x } )
6	5	eleq2d	|- ( K e. D -> ( P e. A <-> P e. { x e. B | .0. C x } ) )
7		breq2	|- ( x = P -> ( .0. C x <-> .0. C P ) )
8	7	elrab	|- ( P e. { x e. B | .0. C x } <-> ( P e. B /\ .0. C P ) )
9	6 8	bitrdi	|- ( K e. D -> ( P e. A <-> ( P e. B /\ .0. C P ) ) )