Theorem lt0neg2

Description: Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004)

	Ref	Expression
Assertion	lt0neg2	⊢ ( 𝐴 ∈ ℝ → ( 0 < 𝐴 ↔ - 𝐴 < 0 ) )

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2		ltneg	⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝐴 ↔ - 𝐴 < - 0 ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℝ → ( 0 < 𝐴 ↔ - 𝐴 < - 0 ) )
4		neg0	⊢ - 0 = 0
5	4	breq2i	⊢ ( - 𝐴 < - 0 ↔ - 𝐴 < 0 )
6	3 5	bitrdi	⊢ ( 𝐴 ∈ ℝ → ( 0 < 𝐴 ↔ - 𝐴 < 0 ) )