Theorem negne0bi

Description: A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999)

	Ref	Expression
Hypothesis	negidi.1	⊢ 𝐴 ∈ ℂ
Assertion	negne0bi	⊢ ( 𝐴 ≠ 0 ↔ - 𝐴 ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		negidi.1	⊢ 𝐴 ∈ ℂ
2		negeq0	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 = 0 ↔ - 𝐴 = 0 ) )
3	1 2	ax-mp	⊢ ( 𝐴 = 0 ↔ - 𝐴 = 0 )
4	3	necon3bii	⊢ ( 𝐴 ≠ 0 ↔ - 𝐴 ≠ 0 )