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

Metamath Proof Explorer

Theorem lt0neg2d

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

Ref Expression
Hypothesis leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
Assertion lt0neg2d ( 𝜑 → ( 0 < 𝐴 ↔ - 𝐴 < 0 ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 lt0neg2 ⊢ ( 𝐴 ∈ ℝ → ( 0 < 𝐴 ↔ - 𝐴 < 0 ) )
3 1 2 syl ⊢ ( 𝜑 → ( 0 < 𝐴 ↔ - 𝐴 < 0 ) )