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 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ≠ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) )

Proof

Step	Hyp	Ref	Expression
1		atsseq	⊢ ( ( 𝐵 ∈ HAtoms ∧ 𝐴 ∈ HAtoms ) → ( 𝐵 ⊆ 𝐴 ↔ 𝐵 = 𝐴 ) )
2		eqcom	⊢ ( 𝐵 = 𝐴 ↔ 𝐴 = 𝐵 )
3	1 2	bitrdi	⊢ ( ( 𝐵 ∈ HAtoms ∧ 𝐴 ∈ HAtoms ) → ( 𝐵 ⊆ 𝐴 ↔ 𝐴 = 𝐵 ) )
4	3	ancoms	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( 𝐵 ⊆ 𝐴 ↔ 𝐴 = 𝐵 ) )
5	4	necon3bbid	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ≠ 𝐵 ) )
6		atelch	⊢ ( 𝐴 ∈ HAtoms → 𝐴 ∈ Cℋ )
7		atnssm0	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) )
8	6 7	sylan	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) )
9	5 8	bitr3d	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ≠ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 0ℋ ) )