Dependent Subset of Independent Set Union Singleton Contains Singleton

Theorem

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

Let $X$ be an independent subset of $M$.

Let $x \in S$.

Let $C$ be a dependent subset of $M$ such that:

$C \subseteq X \cup \set x$.

Then:

$x \in C$

Proof

From the contrapositive statement of Superset of Dependent Set is Dependent:

$C \not \subseteq X$

From the contrapositive statement of Set Difference with Superset is Empty Set:

$C \setminus X \ne \O$

From Set Difference over Subset:

$C \setminus X \subseteq \paren{X \cup \set x} \setminus X = \set x$

From Power Set of Singleton:

$C \setminus X = \set x$

From Set Difference is Subset:

$\set x \subseteq C$

From Singleton of Element is Subset:

$x \in C$

$\blacksquare$