Theorem xle0neg1

Description: Extended real version of le0neg1 . (Contributed by Mario Carneiro, 9-Sep-2015)

	Ref	Expression
Assertion	xle0neg1	$$⊢A\in {ℝ}^{*}\to \left(A\le 0\leftrightarrow 0\le -A\right)$$

Proof

Step	Hyp	Ref	Expression
1		0xr	$$⊢0\in {ℝ}^{*}$$
2		xleneg	$$⊢\left(A\in {ℝ}^{*}\wedge 0\in {ℝ}^{*}\right)\to \left(A\le 0\leftrightarrow -0\le -A\right)$$
3	1 2	mpan2	$$⊢A\in {ℝ}^{*}\to \left(A\le 0\leftrightarrow -0\le -A\right)$$
4		xneg0	$$⊢-0=0$$
5	4	breq1i	$$⊢-0\le -A\leftrightarrow 0\le -A$$
6	3 5	syl6bb	$$⊢A\in {ℝ}^{*}\to \left(A\le 0\leftrightarrow 0\le -A\right)$$