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	$$⊢\phi \to A\in ℝ$$
Assertion	lt0neg1d	$$⊢\phi \to \left(A<0\leftrightarrow 0<-A\right)$$

Proof

Step	Hyp	Ref	Expression
1		leidd.1	$$⊢\phi \to A\in ℝ$$
2		lt0neg1	$$⊢A\in ℝ\to \left(A<0\leftrightarrow 0<-A\right)$$
3	1 2	syl	$$⊢\phi \to \left(A<0\leftrightarrow 0<-A\right)$$