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Metamath Proof Explorer

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 ) )