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| Mirrors > Home > MPE Home > Th. List > gtned | Structured version Visualization version GIF version | ||
| Description: 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| ltned.2 | ⊢ (𝜑 → 𝐴 < 𝐵) |
| Ref | Expression |
|---|---|
| gtned | ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ltd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | ltned.2 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
| 3 | ltne 11358 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
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