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 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) )

Proof

Step	Hyp	Ref	Expression
1		chcv1	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )
2		cvp	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ( 𝐴 ∩ 𝐵 ) = 0ℋ ↔ 𝐴 ⋖ℋ ( 𝐴 ∨ℋ 𝐵 ) ) )
3	1 2	bitr4d	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) )