Step |
Hyp |
Ref |
Expression |
1 |
|
cvnbtwn |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → ¬ ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ) ) |
2 |
|
iman |
⊢ ( ( ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵 ) → 𝐶 = 𝐵 ) ↔ ¬ ( ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵 ) ∧ ¬ 𝐶 = 𝐵 ) ) |
3 |
|
anass |
⊢ ( ( ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵 ) ∧ ¬ 𝐶 = 𝐵 ) ↔ ( 𝐴 ⊊ 𝐶 ∧ ( 𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵 ) ) ) |
4 |
|
dfpss2 |
⊢ ( 𝐶 ⊊ 𝐵 ↔ ( 𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵 ) ) |
5 |
4
|
anbi2i |
⊢ ( ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ↔ ( 𝐴 ⊊ 𝐶 ∧ ( 𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵 ) ) ) |
6 |
3 5
|
bitr4i |
⊢ ( ( ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵 ) ∧ ¬ 𝐶 = 𝐵 ) ↔ ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ) |
7 |
6
|
notbii |
⊢ ( ¬ ( ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵 ) ∧ ¬ 𝐶 = 𝐵 ) ↔ ¬ ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ) |
8 |
2 7
|
bitr2i |
⊢ ( ¬ ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵 ) ↔ ( ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵 ) → 𝐶 = 𝐵 ) ) |
9 |
1 8
|
syl6ib |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( 𝐴 ⋖ℋ 𝐵 → ( ( 𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵 ) → 𝐶 = 𝐵 ) ) ) |