Step |
Hyp |
Ref |
Expression |
1 |
|
ltnelicc.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
ltnelicc.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
3 |
|
ltnelicc.c |
⊢ ( 𝜑 → 𝐶 ∈ ℝ* ) |
4 |
|
ltnelicc.clta |
⊢ ( 𝜑 → 𝐶 < 𝐴 ) |
5 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
6 |
|
xrltnle |
⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶 ) ) |
7 |
3 5 6
|
syl2anc |
⊢ ( 𝜑 → ( 𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶 ) ) |
8 |
4 7
|
mpbid |
⊢ ( 𝜑 → ¬ 𝐴 ≤ 𝐶 ) |
9 |
8
|
intnanrd |
⊢ ( 𝜑 → ¬ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) |
10 |
|
elicc4 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) ) |
11 |
5 2 3 10
|
syl3anc |
⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) ) |
12 |
9 11
|
mtbird |
⊢ ( 𝜑 → ¬ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |