Theorem lesub0i

Description: Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Hypotheses	lt2.1	$$⊢A\in ℝ$$
	lt2.2	$$⊢B\in ℝ$$
Assertion	lesub0i	$$⊢\left(0\le A\wedge B\le B-A\right)\leftrightarrow A=0$$

Proof

Step	Hyp	Ref	Expression
1		lt2.1	$$⊢A\in ℝ$$
2		lt2.2	$$⊢B\in ℝ$$
3		lesub0	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(\left(0\le A\wedge B\le B-A\right)\leftrightarrow A=0\right)$$
4	1 2 3	mp2an	$$⊢\left(0\le A\wedge B\le B-A\right)\leftrightarrow A=0$$