Step |
Hyp |
Ref |
Expression |
1 |
|
atne0 |
⊢ ( 𝐴 ∈ HAtoms → 𝐴 ≠ 0ℋ ) |
2 |
1
|
ad2antrr |
⊢ ( ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ≠ 0ℋ ) |
3 |
|
atelch |
⊢ ( 𝐴 ∈ HAtoms → 𝐴 ∈ Cℋ ) |
4 |
|
atss |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐴 = 0ℋ ) ) ) |
5 |
3 4
|
sylan |
⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐴 = 0ℋ ) ) ) |
6 |
5
|
imp |
⊢ ( ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 = 𝐵 ∨ 𝐴 = 0ℋ ) ) |
7 |
6
|
ord |
⊢ ( ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) ∧ 𝐴 ⊆ 𝐵 ) → ( ¬ 𝐴 = 𝐵 → 𝐴 = 0ℋ ) ) |
8 |
7
|
necon1ad |
⊢ ( ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ≠ 0ℋ → 𝐴 = 𝐵 ) ) |
9 |
2 8
|
mpd |
⊢ ( ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 = 𝐵 ) |
10 |
9
|
ex |
⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⊆ 𝐵 → 𝐴 = 𝐵 ) ) |
11 |
|
eqimss |
⊢ ( 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵 ) |
12 |
10 11
|
impbid1 |
⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⊆ 𝐵 ↔ 𝐴 = 𝐵 ) ) |