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

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

Let $\sigma: \powerset S \to \powerset S$ denote the closure operator of $M$.

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

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

Necessary Condition
Let $x$ and $y$ be parallel.

By definition of parallel:
 * $\set x$ and $\set y$ are independent
 * $\set {x, y}$ is dependent

By definition of a loop:
 * $x$ and $y$ are not loops

From Lemma:
 * $x \in \map \sigma {\set y}$
 * $y \in \map \sigma {\set x}$

It has been shown that conditions $(1), (2)$ and $(3)$ above hold.

Sufficient Condition
Let conditions $(1), (2)$ and $(3)$ above hold.

By definition of a loop:
 * $\set x$ and $\set y$ are independent

From Lemma:
 * $\set {y, x} \notin \mathscr I$

It follows that $x$ is parallel to $y$ by definition.