Theorem leneg

Description: Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	leneg	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\le B\leftrightarrow -B\le -A\right)$$

Proof

Step	Hyp	Ref	Expression
1		0re	$$⊢0\in ℝ$$
2		lesub2	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge 0\in ℝ\right)\to \left(A\le B\leftrightarrow 0-B\le 0-A\right)$$
3	1 2	mp3an3	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\le B\leftrightarrow 0-B\le 0-A\right)$$
4		df-neg	$$⊢-B=0-B$$
5		df-neg	$$⊢-A=0-A$$
6	4 5	breq12i	$$⊢-B\le -A\leftrightarrow 0-B\le 0-A$$
7	3 6	syl6bbr	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\le B\leftrightarrow -B\le -A\right)$$