Theorem atnssm0

Description: The meet of a Hilbert lattice element and an incomparable atom is the zero subspace. (Contributed by NM, 30-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	atnssm0	|- ( ( A e. CH /\ B e. HAtoms ) -> ( -. B C_ A <-> ( A i^i B ) = 0H ) )

Proof

Step	Hyp	Ref	Expression
1		chcv1	|- ( ( A e. CH /\ B e. HAtoms ) -> ( -. B C_ A <-> A
2		cvp	|- ( ( A e. CH /\ B e. HAtoms ) -> ( ( A i^i B ) = 0H <-> A
3	1 2	bitr4d	|- ( ( A e. CH /\ B e. HAtoms ) -> ( -. B C_ A <-> ( A i^i B ) = 0H ) )