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 )