Step |
Hyp |
Ref |
Expression |
1 |
|
atelch |
⊢ ( 𝐵 ∈ HAtoms → 𝐵 ∈ Cℋ ) |
2 |
|
cvpss |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵 ) ) |
3 |
1 2
|
sylan2 |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵 ) ) |
4 |
|
ch0le |
⊢ ( 𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴 ) |
5 |
4
|
adantr |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → 0ℋ ⊆ 𝐴 ) |
6 |
3 5
|
jctild |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⋖ℋ 𝐵 → ( 0ℋ ⊆ 𝐴 ∧ 𝐴 ⊊ 𝐵 ) ) ) |
7 |
|
atcv0 |
⊢ ( 𝐵 ∈ HAtoms → 0ℋ ⋖ℋ 𝐵 ) |
8 |
7
|
adantr |
⊢ ( ( 𝐵 ∈ HAtoms ∧ 𝐴 ∈ Cℋ ) → 0ℋ ⋖ℋ 𝐵 ) |
9 |
|
h0elch |
⊢ 0ℋ ∈ Cℋ |
10 |
|
cvnbtwn3 |
⊢ ( ( 0ℋ ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → ( 0ℋ ⋖ℋ 𝐵 → ( ( 0ℋ ⊆ 𝐴 ∧ 𝐴 ⊊ 𝐵 ) → 𝐴 = 0ℋ ) ) ) |
11 |
9 10
|
mp3an1 |
⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → ( 0ℋ ⋖ℋ 𝐵 → ( ( 0ℋ ⊆ 𝐴 ∧ 𝐴 ⊊ 𝐵 ) → 𝐴 = 0ℋ ) ) ) |
12 |
1 11
|
sylan |
⊢ ( ( 𝐵 ∈ HAtoms ∧ 𝐴 ∈ Cℋ ) → ( 0ℋ ⋖ℋ 𝐵 → ( ( 0ℋ ⊆ 𝐴 ∧ 𝐴 ⊊ 𝐵 ) → 𝐴 = 0ℋ ) ) ) |
13 |
8 12
|
mpd |
⊢ ( ( 𝐵 ∈ HAtoms ∧ 𝐴 ∈ Cℋ ) → ( ( 0ℋ ⊆ 𝐴 ∧ 𝐴 ⊊ 𝐵 ) → 𝐴 = 0ℋ ) ) |
14 |
13
|
ancoms |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( ( 0ℋ ⊆ 𝐴 ∧ 𝐴 ⊊ 𝐵 ) → 𝐴 = 0ℋ ) ) |
15 |
6 14
|
syld |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⋖ℋ 𝐵 → 𝐴 = 0ℋ ) ) |
16 |
|
breq1 |
⊢ ( 𝐴 = 0ℋ → ( 𝐴 ⋖ℋ 𝐵 ↔ 0ℋ ⋖ℋ 𝐵 ) ) |
17 |
7 16
|
syl5ibrcom |
⊢ ( 𝐵 ∈ HAtoms → ( 𝐴 = 0ℋ → 𝐴 ⋖ℋ 𝐵 ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 = 0ℋ → 𝐴 ⋖ℋ 𝐵 ) ) |
19 |
15 18
|
impbid |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⋖ℋ 𝐵 ↔ 𝐴 = 0ℋ ) ) |