Theorem xlt0neg2

Description: Extended real version of lt0neg2 . (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xlt0neg2	$$⊢A\in {ℝ}^{*}\to \left(0<A\leftrightarrow -A<0\right)$$

Proof

Step	Hyp	Ref	Expression
1		0xr	$$⊢0\in {ℝ}^{*}$$
2		xltneg	$$⊢\left(0\in {ℝ}^{*}\wedge A\in {ℝ}^{*}\right)\to \left(0<A\leftrightarrow -A<-0\right)$$
3	1 2	mpan	$$⊢A\in {ℝ}^{*}\to \left(0<A\leftrightarrow -A<-0\right)$$
4		xneg0	$$⊢-0=0$$
5	4	breq2i	$$⊢-A<-0\leftrightarrow -A<0$$
6	3 5	bitrdi	$$⊢A\in {ℝ}^{*}\to \left(0<A\leftrightarrow -A<0\right)$$