Set with Two Parallel Elements is Dependent

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

Let $A \subseteq S$.

Let $x, y \in S$.

Let $x, y$ be parallel elements.

If $x, y \in A$ then $A$ is dependent.

Proof
Let $x, y \in A$. From Doubleton of Elements is Subset:
 * $\set{x, y} \subseteq A$

By the definition of parallel elements:
 * $\set {x, y}$ is dependent

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