Metamath Proof Explorer

Theorem atne0

Description: An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002) (New usage is discouraged.)

		Ref	Expression
	Assertion	atne0	\|- ( A e. HAtoms -> A =/= 0H )

Step	Hyp	Ref	Expression
1		elat2	\|- ( A e. HAtoms <-> ( A e. CH /\ ( A =/= 0H /\ A. x e. CH ( x C_ A -> ( x = A \/ x = 0H ) ) ) ) )
2		simprl	\|- ( ( A e. CH /\ ( A =/= 0H /\ A. x e. CH ( x C_ A -> ( x = A \/ x = 0H ) ) ) ) -> A =/= 0H )
3	1 2	sylbi	\|- ( A e. HAtoms -> A =/= 0H )

Step

Hyp

Ref

Expression

 |-  ( A e. HAtoms <-> ( A e. CH /\ ( A =/= 0H /\ A. x e. CH ( x C_ A -> ( x = A \/ x = 0H ) ) ) ) )

 |-  ( ( A e. CH /\ ( A =/= 0H /\ A. x e. CH ( x C_ A -> ( x = A \/ x = 0H ) ) ) ) -> A =/= 0H )

1 2

 |-  ( A e. HAtoms -> A =/= 0H )