Integer Divisor is Equivalent to Subset of Ideal

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

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

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

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

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

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

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