Elements in Integral Domain are Associates iff Principal Ideals are Equal

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

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

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

Then:
 * $x$ and $y$ are associates $\ideal x = \ideal y$

Proof
Let $x \cong y$ denote that $x$ and $y$ are associates.

Then: