Superset of Dependent Set is Dependent/Corollary

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

Let $x \in A$.

If $x$ is a loop then $A$ is dependent.

Proof
Let $x$ be a loop.

By definition of a loop:
 * $\set x \notin \mathscr I$

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

From Singleton of Element is Subset:
 * $\set x \subseteq A$

From Superset of Dependent Set is Dependent:
 * $A$ is a dependent subset