Step |
Hyp |
Ref |
Expression |
1 |
|
df-ss |
⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 𝐴 ) |
2 |
1
|
biimpi |
⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐵 ) = 𝐴 ) |
3 |
2
|
eleq1d |
⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝐴 ∩ 𝐵 ) ∈ HAtoms ↔ 𝐴 ∈ HAtoms ) ) |
4 |
3
|
biimprcd |
⊢ ( 𝐴 ∈ HAtoms → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐵 ) ∈ HAtoms ) ) |
5 |
4
|
adantr |
⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐵 ) ∈ HAtoms ) ) |
6 |
|
incom |
⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 ) |
7 |
6
|
eleq1i |
⊢ ( ( 𝐴 ∩ 𝐵 ) ∈ HAtoms ↔ ( 𝐵 ∩ 𝐴 ) ∈ HAtoms ) |
8 |
|
atne0 |
⊢ ( ( 𝐵 ∩ 𝐴 ) ∈ HAtoms → ( 𝐵 ∩ 𝐴 ) ≠ 0ℋ ) |
9 |
8
|
neneqd |
⊢ ( ( 𝐵 ∩ 𝐴 ) ∈ HAtoms → ¬ ( 𝐵 ∩ 𝐴 ) = 0ℋ ) |
10 |
7 9
|
sylbi |
⊢ ( ( 𝐴 ∩ 𝐵 ) ∈ HAtoms → ¬ ( 𝐵 ∩ 𝐴 ) = 0ℋ ) |
11 |
|
atnssm0 |
⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ HAtoms ) → ( ¬ 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐴 ) = 0ℋ ) ) |
12 |
11
|
ancoms |
⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( ¬ 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐴 ) = 0ℋ ) ) |
13 |
12
|
biimpd |
⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( ¬ 𝐴 ⊆ 𝐵 → ( 𝐵 ∩ 𝐴 ) = 0ℋ ) ) |
14 |
13
|
con1d |
⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( ¬ ( 𝐵 ∩ 𝐴 ) = 0ℋ → 𝐴 ⊆ 𝐵 ) ) |
15 |
10 14
|
syl5 |
⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( ( 𝐴 ∩ 𝐵 ) ∈ HAtoms → 𝐴 ⊆ 𝐵 ) ) |
16 |
5 15
|
impbid |
⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) ∈ HAtoms ) ) |