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 ) ) |