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Metamath Proof Explorer

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	$⊢ A \in ℝ^{*} \to (A \in ℝ \leftrightarrow - A \in ℝ)$

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