Element in Integral Domain is Divisor iff Principal Ideal is Superset

Theorem
Let $\struct {D, +, \circ}$ be an integral domain.

Let $\ideal x$ denote the principal ideal of $D$ generated by $x$.

Let $x, y \in \struct {D, +, \circ}$.

Then:
 * $x \divides y \iff \ideal y \subseteq \ideal x$

where $x \divides y$ denotes that $x$ is a divisor of $y$.

Proof
Let that $x \divides y$.

Then by definition of divisor:

Conversely:

So:
 * $x \divides y \iff \ideal y \subseteq \ideal x$