Theorem atnemeq0

Description: The meet of distinct atoms is the zero subspace. (Contributed by NM, 25-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	atnemeq0	|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( A =/= B <-> ( A i^i B ) = 0H ) )

Proof

Step	Hyp	Ref	Expression
1		atsseq	|- ( ( B e. HAtoms /\ A e. HAtoms ) -> ( B C_ A <-> B = A ) )
2		eqcom	|- ( B = A <-> A = B )
3	1 2	bitrdi	|- ( ( B e. HAtoms /\ A e. HAtoms ) -> ( B C_ A <-> A = B ) )
4	3	ancoms	|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( B C_ A <-> A = B ) )
5	4	necon3bbid	|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( -. B C_ A <-> A =/= B ) )
6		atelch	|- ( A e. HAtoms -> A e. CH )
7		atnssm0	|- ( ( A e. CH /\ B e. HAtoms ) -> ( -. B C_ A <-> ( A i^i B ) = 0H ) )
8	6 7	sylan	|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( -. B C_ A <-> ( A i^i B ) = 0H ) )
9	5 8	bitr3d	|- ( ( A e. HAtoms /\ B e. HAtoms ) -> ( A =/= B <-> ( A i^i B ) = 0H ) )