Theorem lt0neg2

Description: Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004)

	Ref	Expression
Assertion	lt0neg2	$$⊢A\in ℝ\to \left(0<A\leftrightarrow -A<0\right)$$

Proof

Step	Hyp	Ref	Expression
1		0re	$$⊢0\in ℝ$$
2		ltneg	$$⊢\left(0\in ℝ\wedge A\in ℝ\right)\to \left(0<A\leftrightarrow -A<-0\right)$$
3	1 2	mpan	$$⊢A\in ℝ\to \left(0<A\leftrightarrow -A<-0\right)$$
4		neg0	$$⊢-0=0$$
5	4	breq2i	$$⊢-A<-0\leftrightarrow -A<0$$
6	3 5	bitrdi	$$⊢A\in ℝ\to \left(0<A\leftrightarrow -A<0\right)$$