Matroid Unique Circuit Property

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

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

Let $x \in S$ such that:
 * $X \cup \set x$ is a dependent subset of $M$.

Then there exists a unique circuit $C$ such that:
 * $C \subseteq X \cup \set x$

Proof
From Element of Finite Ordered Set is Between Maximal and Minimal Elements:
 * $\exists C \in \powerset S \setminus \mathscr I : C \subseteq X \cup \set x$ and $C$ is minimal in $\struct {\powerset S \setminus \mathscr I, \subseteq}$.