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Metamath Proof Explorer

Theorem negne0d

Description: The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses negidd.1 ⊢ φ → A ∈ ℂ
negne0d.2 ⊢ φ → A ≠ 0
Assertion negne0d ⊢ φ → − A ≠ 0

Proof

Step Hyp Ref Expression
1 negidd.1 ⊢ φ → A ∈ ℂ
2 negne0d.2 ⊢ φ → A ≠ 0
3 1 negne0bd ⊢ φ → A ≠ 0 ↔ − A ≠ 0
4 2 3 mpbid ⊢ φ → − A ≠ 0