Theorem relt0neg1

Description: Comparison of a real and its negative to zero. Compare lt0neg1 . (Contributed by SN, 13-Feb-2024)

	Ref	Expression
Assertion	relt0neg1	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < 0 ↔ 0 < ( 0 −ℝ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2		reposdif	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐴 < 0 ↔ 0 < ( 0 −ℝ 𝐴 ) ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < 0 ↔ 0 < ( 0 −ℝ 𝐴 ) ) )