Theorem le0neg1

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

	Ref	Expression
Assertion	le0neg1	$$⊢A\in ℝ\to \left(A\le 0\leftrightarrow 0\le -A\right)$$

Proof

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