Theorem lenegi

Description: Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999)

	Ref	Expression
Hypotheses	lt2.1	$$⊢A\in ℝ$$
	lt2.2	$$⊢B\in ℝ$$
Assertion	lenegi	$$⊢A\le B\leftrightarrow -B\le -A$$

Proof

Step	Hyp	Ref	Expression
1		lt2.1	$$⊢A\in ℝ$$
2		lt2.2	$$⊢B\in ℝ$$
3		leneg	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\le B\leftrightarrow -B\le -A\right)$$
4	1 2 3	mp2an	$$⊢A\le B\leftrightarrow -B\le -A$$