Union of Matroid Base with Element of Complement is Dependent

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

Let $B \subseteq S$ be a base of $M$.

Let $x \in S \setminus B$.

Then:
 * $B \cup \set x$ is a dependent superset of $B$

Proof
From Set is Subset of Union:
 * $B \subseteq B \cup \set x$

Because $x \in B \cup \set x$ and $x \notin B$:
 * $B \ne B \cup \set x$

Hence:
 * $B \subsetneq B \cup \set x$

By definition of base:
 * $B$ is a maximal independent subset

Hence:
 * $B \cup \set x \notin \mathscr I$