Theorem elatcv0

Description: A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	elatcv0	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to \left(A\in \mathrm{HAtoms}\leftrightarrow 0_{\mathscr{H}}{⋖}_{\mathscr{H}}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ela	$$⊢A\in \mathrm{HAtoms}\leftrightarrow \left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge 0_{\mathscr{H}}{⋖}_{\mathscr{H}}A\right)$$
2	1	baib	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}\to \left(A\in \mathrm{HAtoms}\leftrightarrow 0_{\mathscr{H}}{⋖}_{\mathscr{H}}A\right)$$