Step |
Hyp |
Ref |
Expression |
1 |
|
xrlttri5d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
2 |
|
xrlttri5d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
3 |
|
xrlttri5d.aneb |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
4 |
|
xrlttri5d.nlt |
⊢ ( 𝜑 → ¬ 𝐵 < 𝐴 ) |
5 |
3
|
neneqd |
⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 ) |
6 |
|
xrlttri3 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) ) |
7 |
1 2 6
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) ) |
8 |
5 7
|
mtbid |
⊢ ( 𝜑 → ¬ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) |
9 |
|
oran |
⊢ ( ( 𝐴 < 𝐵 ∨ 𝐵 < 𝐴 ) ↔ ¬ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) |
10 |
8 9
|
sylibr |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ∨ 𝐵 < 𝐴 ) ) |
11 |
10 4
|
jca |
⊢ ( 𝜑 → ( ( 𝐴 < 𝐵 ∨ 𝐵 < 𝐴 ) ∧ ¬ 𝐵 < 𝐴 ) ) |
12 |
|
pm5.61 |
⊢ ( ( ( 𝐴 < 𝐵 ∨ 𝐵 < 𝐴 ) ∧ ¬ 𝐵 < 𝐴 ) ↔ ( 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) |
13 |
11 12
|
sylib |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) |
14 |
13
|
simpld |
⊢ ( 𝜑 → 𝐴 < 𝐵 ) |