Element Depends on Independent Set iff Union with Singleton is Dependent

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

Let $X \in \mathscr I$.

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

Then:
 * $x \in \map \sigma X$ $X \cup \set x$ is dependent.

Necessary Condition
Let $x \in \map \sigma X$.

By definition of the closure:
 * $x$ depends on $X$.

From Union with Disjoint Singleton is Dependent if Element Depends on Subset:
 * $X \cup \set x$ is dependent.

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

Lemma
From Max Operation Yields Supremum of Operands:
 * $\map \rho {X \cup \set x} = \max \set {\size A : A \subseteq X \cup \set x \land A \in \mathscr I} \le \size X$

By assumption:
 * $X \in \mathscr I$

From Max Operation Yields Supremum of Operands:
 * $\size X \le \map \rho {X \cup \set x}$

Thus:
 * $\size X = \map \rho {X \cup \set x}$

From Rank of Independent Subset Equals Cardinality:
 * $\map \rho X = \size X$

Thus:
 * $\map \rho X = \map \rho {X \cup \set x}$

It follows that $x$ depends on $X$ by definition.

So:
 * $x \in \map \sigma X$