Step |
Hyp |
Ref |
Expression |
1 |
|
xrlttri |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
2 |
|
ioran |
⊢ ( ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ↔ ( ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) |
3 |
2
|
biancomi |
⊢ ( ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ↔ ( ¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵 ) ) |
4 |
1 3
|
bitrdi |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ( ¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵 ) ) ) |
5 |
|
xrlenlt |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) ) |
6 |
|
nesym |
⊢ ( 𝐵 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐵 ) |
7 |
6
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 ≠ 𝐴 ↔ ¬ 𝐴 = 𝐵 ) ) |
8 |
5 7
|
anbi12d |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) ↔ ( ¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵 ) ) ) |
9 |
4 8
|
bitr4d |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≠ 𝐴 ) ) ) |