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 )