Theorem negeq0

Description: A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	negeq0	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 = 0 ↔ - 𝐴 = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		0cn	⊢ 0 ∈ ℂ
2		neg11	⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ) → ( - 𝐴 = - 0 ↔ 𝐴 = 0 ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 = - 0 ↔ 𝐴 = 0 ) )
4		neg0	⊢ - 0 = 0
5	4	eqeq2i	⊢ ( - 𝐴 = - 0 ↔ - 𝐴 = 0 )
6	3 5	bitr3di	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 = 0 ↔ - 𝐴 = 0 ) )