Metamath Proof Explorer


Theorem le0neg1d

Description: Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypothesis leidd.1 ( 𝜑𝐴 ∈ ℝ )
Assertion le0neg1d ( 𝜑 → ( 𝐴 ≤ 0 ↔ 0 ≤ - 𝐴 ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ( 𝜑𝐴 ∈ ℝ )
2 le0neg1 ( 𝐴 ∈ ℝ → ( 𝐴 ≤ 0 ↔ 0 ≤ - 𝐴 ) )
3 1 2 syl ( 𝜑 → ( 𝐴 ≤ 0 ↔ 0 ≤ - 𝐴 ) )