Description: Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | eliood.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) | |
eliood.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) | ||
eliood.3 | ⊢ ( 𝜑 → 𝐶 ∈ ℝ ) | ||
eliood.4 | ⊢ ( 𝜑 → 𝐴 < 𝐶 ) | ||
eliood.5 | ⊢ ( 𝜑 → 𝐶 < 𝐵 ) | ||
Assertion | eliood | ⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliood.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) | |
2 | eliood.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) | |
3 | eliood.3 | ⊢ ( 𝜑 → 𝐶 ∈ ℝ ) | |
4 | eliood.4 | ⊢ ( 𝜑 → 𝐴 < 𝐶 ) | |
5 | eliood.5 | ⊢ ( 𝜑 → 𝐶 < 𝐵 ) | |
6 | elioo2 | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) ) | |
7 | 1 2 6 | syl2anc | ⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) ) |
8 | 3 4 5 7 | mpbir3and | ⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) |