Distinct Matroid Elements are Parallel iff Each is in Closure of Other

Theorem
Let $M = \struct{S, \mathscr I}$ be a matroid.

Let $x, y \in S : x \ne y$.

Then $x$ is parallel to $y$ :
 * $(1)$ $x$ and $y$ are not loops
 * $(2)$ $x \in \map \sigma {\set y}$
 * $(3)$ $y \in \map \sigma {\set x}$