Union with Disjoint Singleton is Dependent if Element Depends on Subset

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

Let $A \subseteq S$.

Let $x \in S : x \notin A$.

Let $x$ depend on $A$.

Then $A \cup \set x$ is dependent

Proof
We proceed by Proof by Contraposition.

Let $A \cup \set x$ be independent.

By matroid axiom $(\text I 2)$:
 * $A$ is independent

We have:

Then $x$ does not depend on $A$ by definition.

The theorem holds by the Rule of Transposition.