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| Mirrors > Home > MPE Home > Th. List > subne0d | Structured version Visualization version GIF version | ||
| Description: Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017.) |
| Ref | Expression |
|---|---|
| negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| subne0d.3 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| Ref | Expression |
|---|---|
| subne0d | ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | subne0d.3 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
| 2 | negidd.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 3 | pncand.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 4 | subeq0 11535 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) | |
| 5 | 2, 3, 4 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
| 6 | 5 | necon3bid 2985 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) ≠ 0 ↔ 𝐴 ≠ 𝐵)) |
| 7 | 1, 6 | mpbird 257 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0) |
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