Element Depends on Independent Set iff Union with Singleton is Dependent/Lemma

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

Let $X \in \mathscr I$.

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

Let $X \cup \set x$ be dependent.

Let $A \in \mathscr I$ such that $A \subseteq X \cup \set x$.

Then:
 * $\size A \le \size X$

Case 1: $x \in A$
Let $x \in A$.

We have:


 * $A \setminus \set x = X$
 * $A \setminus \set x = X$

Then:

So:
 * $X \cup \set x$ is independent.

This contradicts:
 * $X \cup \set x$ is dependent.

So:
 * $A \setminus \set x \subsetneq X$

Then:

So:
 * $\size A \le \size X$

Case 2: $x \notin A$
Let $x \notin A$.

Then:

From Intersection with Subset is Subset:
 * $A \subseteq X$

From Cardinality of Subset of Finite Set:
 * $\size A \le \size X$

In either case:
 * $\size A \le \size X$