Theorem relt0neg1

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

	Ref	Expression
Assertion	relt0neg1	$$⊢A\in ℝ\to \left(A<0\leftrightarrow 0<0{-}_{ℝ}A\right)$$

Proof

Step	Hyp	Ref	Expression
1		0re	$$⊢0\in ℝ$$
2		reposdif	$$⊢\left(A\in ℝ\wedge 0\in ℝ\right)\to \left(A<0\leftrightarrow 0<0{-}_{ℝ}A\right)$$
3	1 2	mpan2	$$⊢A\in ℝ\to \left(A<0\leftrightarrow 0<0{-}_{ℝ}A\right)$$