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	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathrm{HAtoms}\right)\to \left(\neg B\subseteq A\leftrightarrow A\cap B=0_{\mathscr{H}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		chcv1	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathrm{HAtoms}\right)\to \left(\neg B\subseteq A\leftrightarrow A{⋖}_{\mathscr{H}}\left(A{\vee }_{\mathscr{H}}B\right)\right)$$
2		cvp	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathrm{HAtoms}\right)\to \left(A\cap B=0_{\mathscr{H}}\leftrightarrow A{⋖}_{\mathscr{H}}\left(A{\vee }_{\mathscr{H}}B\right)\right)$$
3	1 2	bitr4d	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathrm{HAtoms}\right)\to \left(\neg B\subseteq A\leftrightarrow A\cap B=0_{\mathscr{H}}\right)$$