Theorem lt0neg1

Description: Comparison of a number and its negative to zero. Theorem I.23 of Apostol p. 20. (Contributed by NM, 14-May-1999)

	Ref	Expression
Assertion	lt0neg1	$$⊢A\in ℝ\to \left(A<0\leftrightarrow 0<-A\right)$$

Proof

Step	Hyp	Ref	Expression
1		0re	$$⊢0\in ℝ$$
2		ltneg	$$⊢\left(A\in ℝ\wedge 0\in ℝ\right)\to \left(A<0\leftrightarrow -0<-A\right)$$
3	1 2	mpan2	$$⊢A\in ℝ\to \left(A<0\leftrightarrow -0<-A\right)$$
4		neg0	$$⊢-0=0$$
5	4	breq1i	$$⊢-0<-A\leftrightarrow 0<-A$$
6	3 5	bitrdi	$$⊢A\in ℝ\to \left(A<0\leftrightarrow 0<-A\right)$$