Theorem xnegrecl2d

Description: If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypotheses	xnegrecl2d.1	$$⊢\phi \to A\in {ℝ}^{*}$$
	xnegrecl2d.2	$$⊢\phi \to -A\in ℝ$$
Assertion	xnegrecl2d	$$⊢\phi \to A\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		xnegrecl2d.1	$$⊢\phi \to A\in {ℝ}^{*}$$
2		xnegrecl2d.2	$$⊢\phi \to -A\in ℝ$$
3		xnegrecl2	$$⊢\left(A\in {ℝ}^{*}\wedge -A\in ℝ\right)\to A\in ℝ$$
4	1 2 3	syl2anc	$$⊢\phi \to A\in ℝ$$