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| Mirrors > Home > MPE Home > Th. List > lelttrd | Structured version Visualization version GIF version | ||
| Description: Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.) |
| Ref | Expression |
|---|---|
| ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
| letrd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
| lelttrd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| lelttrd.5 | ⊢ (𝜑 → 𝐵 < 𝐶) |
| Ref | Expression |
|---|---|
| lelttrd | ⊢ (𝜑 → 𝐴 < 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lelttrd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
| 2 | lelttrd.5 | . 2 ⊢ (𝜑 → 𝐵 < 𝐶) | |
| 3 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 4 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 5 | letrd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
| 6 | lelttr 11351 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) | |
| 7 | 3, 4, 5, 6 | syl3anc 1373 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶)) |
| 8 | 1, 2, 7 | mp2and 699 | 1 ⊢ (𝜑 → 𝐴 < 𝐶) |
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