Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliood | Structured version Visualization version GIF version |
Description: Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
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) |
Ref | Expression |
---|---|
eliood | ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliood.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
2 | eliood.4 | . 2 ⊢ (𝜑 → 𝐴 < 𝐶) | |
3 | eliood.5 | . 2 ⊢ (𝜑 → 𝐶 < 𝐵) | |
4 | eliood.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
5 | eliood.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
6 | elioo2 12976 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
7 | 4, 5, 6 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
8 | 1, 2, 3, 7 | mpbir3and 1344 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) |
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