Theorem nnsubi

Description: Subtraction of positive integers. (Contributed by NM, 19-Aug-2001)

	Ref	Expression
Hypotheses	nnsub.1	$$⊢A\in \mathbb{N}$$
	nnsub.2	$$⊢B\in \mathbb{N}$$
Assertion	nnsubi	$$⊢A<B\leftrightarrow B-A\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		nnsub.1	$$⊢A\in \mathbb{N}$$
2		nnsub.2	$$⊢B\in \mathbb{N}$$
3		nnsub	$$⊢\left(A\in \mathbb{N}\wedge B\in \mathbb{N}\right)\to \left(A<B\leftrightarrow B-A\in \mathbb{N}\right)$$
4	1 2 3	mp2an	$$⊢A<B\leftrightarrow B-A\in \mathbb{N}$$