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Metamath Proof Explorer

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 ⊢ φ → A ∈ ℝ *
xnegrecl2d.2 ⊢ φ → − A ∈ ℝ
Assertion xnegrecl2d ⊢ φ → A ∈ ℝ

Proof

Step Hyp Ref Expression
1 xnegrecl2d.1 ⊢ φ → A ∈ ℝ *
2 xnegrecl2d.2 ⊢ φ → − A ∈ ℝ
3 xnegrecl2 ⊢ A ∈ ℝ * ∧ − A ∈ ℝ → A ∈ ℝ
4 1 2 3 syl2anc ⊢ φ → A ∈ ℝ