Distinct Elements are Parallel iff Pair forms Circuit

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

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

Then:
 * $x$ and $y$ are parallel $\set {x, y}$ is a circuit

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

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

Let $A \subseteq \set {x, y}$ be dependent.

Thus:
 * $A \ne \set x, \set y$

By matroid axiom $(\text I 1)$:
 * $\O$ is independent

Thus:
 * $A \ne \O$

From Power Set of Doubleton:
 * $\powerset {\set {x, y} } = \set {\O, \set x, \set y, \set {x, y} }$

Thus:
 * $A = \set {x, y}$

By definition of a minimal set:
 * $\set {x, y}$ is a minimal dependent subset

It follows that $\set {x, y}$ is a circuit by definition.

Sufficient Condition
Let $\set {x, y}$ by a circuit.

By definition of a circuit:
 * $\set {x, y}$ is a minimal dependent subset

By definition of a subset:
 * $\set x, \set y \subseteq \set {x, y}$

By definition of a minimal dependent subset:
 * $\set x, \set y$ are independent subsets

It follows that $x$ and $y$ are parallel by definition.