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Metamath Proof Explorer

Theorem lt0neg1dd

Description: If a number is negative, its negative is positive. (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses lt0neg1dd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
lt0neg1dd.2 ⊢ ( 𝜑 → 𝐴 < 0 )
Assertion lt0neg1dd ( 𝜑 → 0 < - 𝐴 )

Proof

Step Hyp Ref Expression
1 lt0neg1dd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 lt0neg1dd.2 ⊢ ( 𝜑 → 𝐴 < 0 )
3 1 lt0neg1d ⊢ ( 𝜑 → ( 𝐴 < 0 ↔ 0 < - 𝐴 ) )
4 2 3 mpbid ⊢ ( 𝜑 → 0 < - 𝐴 )