Step |
Hyp |
Ref |
Expression |
1 |
|
sltletr |
⊢ ( ( 𝐶 ∈ No ∧ 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐶 <s 𝐴 ∧ 𝐴 ≤s 𝐵 ) → 𝐶 <s 𝐵 ) ) |
2 |
1
|
3coml |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐶 <s 𝐴 ∧ 𝐴 ≤s 𝐵 ) → 𝐶 <s 𝐵 ) ) |
3 |
2
|
expcomd |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 → ( 𝐶 <s 𝐴 → 𝐶 <s 𝐵 ) ) ) |
4 |
3
|
imp |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐴 ≤s 𝐵 ) → ( 𝐶 <s 𝐴 → 𝐶 <s 𝐵 ) ) |
5 |
4
|
con3d |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐴 ≤s 𝐵 ) → ( ¬ 𝐶 <s 𝐵 → ¬ 𝐶 <s 𝐴 ) ) |
6 |
5
|
expimpd |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ≤s 𝐵 ∧ ¬ 𝐶 <s 𝐵 ) → ¬ 𝐶 <s 𝐴 ) ) |
7 |
|
slenlt |
⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵 ) ) |
8 |
7
|
3adant1 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵 ) ) |
9 |
8
|
anbi2d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶 ) ↔ ( 𝐴 ≤s 𝐵 ∧ ¬ 𝐶 <s 𝐵 ) ) ) |
10 |
|
slenlt |
⊢ ( ( 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐴 ) ) |
11 |
10
|
3adant2 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐴 ) ) |
12 |
6 9 11
|
3imtr4d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶 ) → 𝐴 ≤s 𝐶 ) ) |