Theorem lt0neg1d

Description: Comparison of a number and its negative to zero. Theorem I.23 of Apostol p. 20. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypothesis	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
Assertion	lt0neg1d	⊢ ( 𝜑 → ( 𝐴 < 0 ↔ 0 < - 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		lt0neg1	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < 0 ↔ 0 < - 𝐴 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐴 < 0 ↔ 0 < - 𝐴 ) )