Dependent Subset of Independent Set Union Singleton Contains Singleton

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

Let $X \subseteq S$ be an independent subset of $M$.

Let $x \in S$.

Let $C$ be a dependent subset of $M$ such that:
 * $C \subseteq X \cup \set x$.

Then:
 * $x \in C$

Proof
From the contrapositive statement of Superset of Dependent Set is Dependent:
 * $C \not \subseteq X$

From the contrapositive statement of Set Difference with Superset is Empty Set:
 * $C \setminus X \ne \O$

From Set Difference over Subset:
 * $C \setminus X \subseteq \paren{X \cup \set x} \setminus X = \set x$

From Power Set of Singleton:
 * $C \setminus X = \set x$

From Set Difference if Subset:
 * $\set x \subseteq C$

From Singleton of Element is Subset:
 * $x \in C$