User:Leigh.Samphier/Sandbox/Independent Superset of Dependent Set Minus Singleton Doesn't Contain Singleton

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

Let $C$ be a dependent subset of $M$.

Let $x \in S$.

Let $X$ be an independent subset of $M$ such that:
 * $C \setminus \set x \subseteq X$.

Then:
 * $x \notin X$

Proof
We prove the contrapositive statement:
 * $x \in X \implies X$ is a dependent subset.

Let $x \in X$.

We have:

From Superset of Dependent Set is Dependent:
 * $X$ is a dependent subset