Theorem xnegre

Description: An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Assertion	xnegre	$$⊢A\in {ℝ}^{*}\to \left(A\in ℝ\leftrightarrow -A\in ℝ\right)$$

Proof

Step	Hyp	Ref	Expression
1		xnegrecl	$$⊢A\in ℝ\to -A\in ℝ$$
2	1	adantl	$$⊢\left(A\in {ℝ}^{*}\wedge A\in ℝ\right)\to -A\in ℝ$$
3		xnegrecl2	$$⊢\left(A\in {ℝ}^{*}\wedge -A\in ℝ\right)\to A\in ℝ$$
4	2 3	impbida	$$⊢A\in {ℝ}^{*}\to \left(A\in ℝ\leftrightarrow -A\in ℝ\right)$$