# 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

$C \nsubseteq X$
$C \setminus X \ne \O$
$C \setminus X \subseteq \paren {X \cup \set x} \setminus X = \set x$
$C \setminus X = \set x$
$\set x \subseteq C$
$x \in C$

$\blacksquare$