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		elre0re	$$⊢A\in ℝ\to 0\in ℝ$$
2		id	$$⊢A\in ℝ\to A\in ℝ$$
3		reltsub1	$$⊢\left(A\in ℝ\wedge 0\in ℝ\wedge A\in ℝ\right)\to \left(A<0\leftrightarrow A{-}_{ℝ}A<0{-}_{ℝ}A\right)$$
4	1 2 3	mpd3an23	$$⊢A\in ℝ\to \left(A<0\leftrightarrow A{-}_{ℝ}A<0{-}_{ℝ}A\right)$$
5		resubid	$$⊢A\in ℝ\to A{-}_{ℝ}A=0$$
6	5	breq1d	$$⊢A\in ℝ\to \left(A{-}_{ℝ}A<0{-}_{ℝ}A\leftrightarrow 0<0{-}_{ℝ}A\right)$$
7	4 6	bitrd	$$⊢A\in ℝ\to \left(A<0\leftrightarrow 0<0{-}_{ℝ}A\right)$$