Step |
Hyp |
Ref |
Expression |
1 |
|
df-ioo |
⊢ (,) = ( 𝑎 ∈ ℝ* , 𝑏 ∈ ℝ* ↦ { 𝑥 ∈ ℝ* ∣ ( 𝑎 < 𝑥 ∧ 𝑥 < 𝑏 ) } ) |
2 |
|
df-ico |
⊢ [,) = ( 𝑎 ∈ ℝ* , 𝑏 ∈ ℝ* ↦ { 𝑥 ∈ ℝ* ∣ ( 𝑎 ≤ 𝑥 ∧ 𝑥 < 𝑏 ) } ) |
3 |
|
xrlenlt |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵 ) ) |
4 |
1 2 3
|
ixxdisj |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 (,) 𝐵 ) ∩ ( 𝐵 [,) 𝐶 ) ) = ∅ ) |
5 |
4
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) → ( ( 𝐴 (,) 𝐵 ) ∩ ( 𝐵 [,) 𝐶 ) ) = ∅ ) |
6 |
|
xrltletr |
⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝑤 < 𝐶 ) ) |
7 |
|
xrltletr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝑤 ) → 𝐴 < 𝑤 ) ) |
8 |
1 2 3 1 6 7
|
ixxun |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) → ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐵 [,) 𝐶 ) ) = ( 𝐴 (,) 𝐶 ) ) |
9 |
5 8
|
jca |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) → ( ( ( 𝐴 (,) 𝐵 ) ∩ ( 𝐵 [,) 𝐶 ) ) = ∅ ∧ ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐵 [,) 𝐶 ) ) = ( 𝐴 (,) 𝐶 ) ) ) |