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| Mirrors > Home > MPE Home > Th. List > leneltd | Structured version Visualization version GIF version | ||
| Description: 'Less than or equal to' and 'not equals' implies 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
| leltned.3 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| leneltd.4 | ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
| Ref | Expression |
|---|---|
| leneltd | ⊢ (𝜑 → 𝐴 < 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | leneltd.4 | . 2 ⊢ (𝜑 → 𝐵 ≠ 𝐴) | |
| 2 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 3 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 4 | leltned.3 | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
| 5 | 2, 3, 4 | leltned 11414 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴)) |
| 6 | 1, 5 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴 < 𝐵) |
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