Theorem islln4

Description: The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012)

	Ref	Expression
Hypotheses	llnset.b	|- B = ( Base ` K )
	llnset.c	|- C = (
	llnset.a	|- A = ( Atoms ` K )
	llnset.n	|- N = ( LLines ` K )
Assertion	islln4	|- ( ( K e. D /\ X e. B ) -> ( X e. N <-> E. p e. A p C X ) )

Proof

Step	Hyp	Ref	Expression
1		llnset.b	|- B = ( Base ` K )
2		llnset.c	|- C = (
3		llnset.a	|- A = ( Atoms ` K )
4		llnset.n	|- N = ( LLines ` K )
5	1 2 3 4	islln	|- ( K e. D -> ( X e. N <-> ( X e. B /\ E. p e. A p C X ) ) )
6	5	baibd	|- ( ( K e. D /\ X e. B ) -> ( X e. N <-> E. p e. A p C X ) )