Theorem posdifi

Description: Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001)

	Ref	Expression
Hypotheses	lt2.1	$$⊢A\in ℝ$$
	lt2.2	$$⊢B\in ℝ$$
Assertion	posdifi	$$⊢A<B\leftrightarrow 0<B-A$$

Proof

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