Theorem xnegred

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

	Ref	Expression
Hypothesis	xnegred.1	$$⊢\phi \to A\in {ℝ}^{*}$$
Assertion	xnegred	$$⊢\phi \to \left(A\in ℝ\leftrightarrow -A\in ℝ\right)$$

Proof

Step	Hyp	Ref	Expression
1		xnegred.1	$$⊢\phi \to A\in {ℝ}^{*}$$
2		xnegre	$$⊢A\in {ℝ}^{*}\to \left(A\in ℝ\leftrightarrow -A\in ℝ\right)$$
3	1 2	syl	$$⊢\phi \to \left(A\in ℝ\leftrightarrow -A\in ℝ\right)$$