Theorem subge02

Description: Nonnegative subtraction. (Contributed by NM, 27-Jul-2005)

	Ref	Expression
Assertion	subge02	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(0\le B\leftrightarrow A-B\le A\right)$$

Proof

Step	Hyp	Ref	Expression
1		addge01	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(0\le B\leftrightarrow A\le A+B\right)$$
2		lesubadd	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge A\in ℝ\right)\to \left(A-B\le A\leftrightarrow A\le A+B\right)$$
3	2	3anidm13	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A-B\le A\leftrightarrow A\le A+B\right)$$
4	1 3	bitr4d	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(0\le B\leftrightarrow A-B\le A\right)$$