This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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

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