Integer Divisor is Equivalent to Subset of Ideal

Theorem
Let $$\mathbb{Z}$$ be the set of all integers.

Let $$\mathbb{Z}^*_+$$ be the set of strictly positive integers.

Let $$m \in \mathbb{Z}^*_+$$ and let $$n \in \mathbb{Z}$$.

Let $$\left({m}\right)$$ be the principal ideal of $$\mathbb{Z}$$ generated by $$m$$.

Then $$m \backslash n \iff \left({n}\right) \subseteq \left({m}\right)$$.

Proof
By definition, $$m \backslash n \iff n \in \left({m}\right)$$.

But $$n \in \left({m}\right) \iff \left({n}\right) \subseteq \left({m}\right)$$.