Metamath Proof Explorer


Theorem islln

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

Ref Expression
Hypotheses llnset.b B = Base K
llnset.c C = K
llnset.a A = Atoms K
llnset.n N = LLines K
Assertion islln K D X N X B p A p C X

Proof

Step Hyp Ref Expression
1 llnset.b B = Base K
2 llnset.c C = K
3 llnset.a A = Atoms K
4 llnset.n N = LLines K
5 1 2 3 4 llnset K D N = x B | p A p C x
6 5 eleq2d K D X N X x B | p A p C x
7 breq2 x = X p C x p C X
8 7 rexbidv x = X p A p C x p A p C X
9 8 elrab X x B | p A p C x X B p A p C X
10 6 9 bitrdi K D X N X B p A p C X