Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 . (Contributed by FL, 29-May-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | ubioc1 | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ( 𝐴 (,] 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ℝ* ) | |
2 | simp3 | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 < 𝐵 ) | |
3 | xrleid | ⊢ ( 𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵 ) | |
4 | 3 | 3ad2ant2 | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ≤ 𝐵 ) |
5 | elioc1 | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) ) | |
6 | 5 | 3adant3 | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( 𝐵 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) ) |
7 | 1 2 4 6 | mpbir3and | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ( 𝐴 (,] 𝐵 ) ) |