Description: An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021)
Ref | Expression | ||
---|---|---|---|
Hypotheses | iooltubd.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) | |
iooltubd.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) | ||
iooltubd.3 | ⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) | ||
Assertion | iooltubd | ⊢ ( 𝜑 → 𝐶 < 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iooltubd.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) | |
2 | iooltubd.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) | |
3 | iooltubd.3 | ⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) | |
4 | iooltub | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) → 𝐶 < 𝐵 ) | |
5 | 1 2 3 4 | syl3anc | ⊢ ( 𝜑 → 𝐶 < 𝐵 ) |