Theorem le0neg2

Description: Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999)

	Ref	Expression
Assertion	le0neg2	$$⊢A\in ℝ\to \left(0\le A\leftrightarrow -A\le 0\right)$$

Proof

Step	Hyp	Ref	Expression
1		0re	$$⊢0\in ℝ$$
2		leneg	$$⊢\left(0\in ℝ\wedge A\in ℝ\right)\to \left(0\le A\leftrightarrow -A\le -0\right)$$
3	1 2	mpan	$$⊢A\in ℝ\to \left(0\le A\leftrightarrow -A\le -0\right)$$
4		neg0	$$⊢-0=0$$
5	4	breq2i	$$⊢-A\le -0\leftrightarrow -A\le 0$$
6	3 5	syl6bb	$$⊢A\in ℝ\to \left(0\le A\leftrightarrow -A\le 0\right)$$