Element is Loop iff Singleton is Circuit

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

Let $x \in S$.

Then:
 * $x$ is a loop $\set x$ is a circuit

Necessary Condition
Let $x$ be a loop.

By definition of a loop:
 * $\set x$ is a dependent subset of $S$

Let $A \subseteq \set x$ be a dependent subset.

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

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

Then:
 * $A = \set x$

It follows that:
 * $\set x$ is a minimal dependent subset of $S$.

Then $\set x$ be a circuit by definition.

Sufficient Condition
Let $\set x$ be a circuit.

By definition of a circuit:
 * $\set x$ is a minimal dependent subset of $S$.

In particular, $\set x$ is a dependent subset of $S$.

Then $\set x$ is a loop by definition.