Matroid Unique Circuit Property/Proof 1

Proof
From Dependent Subset Contains a Circuit:
 * there exists a circuit $C$ such that $C \subseteq X \cup \set x$

From Dependent Subset of Independent Set Union Singleton Contains Singleton:
 * $x \in C$

$C'$ is circuit of $M$ such that:
 * $C' \ne C$
 * $C' \subseteq X \cup \set x$

From Dependent Subset of Independent Set Union Singleton Contains Singleton:
 * $x \in C'$

Hence:
 * $x \in C \cap C'$

From Equivalence of Definitions of Matroid Circuit Axioms, the set $\mathscr C$ of all circuits satisfies the matroid circuit axiom $(\text C 3)$:

Hence there exists a circuit $C_3$ of $M$:
 * $C_3 \subseteq \paren{C \cup C'} \setminus \set x \subseteq X$

This contradicts the independence of $X$.

The result follows.