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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 11524 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) | |
5 | 2, 3, 4 | syl2anc 582 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
6 | 5 | necon3bid 2982 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) ≠ 0 ↔ 𝐴 ≠ 𝐵)) |
7 | 1, 6 | mpbird 256 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0) |
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