Independent Subset is Contained in Base

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

Let $\mathscr B$ denote the set of all bases of $M$.

Let $A \in \mathscr I$.

Then:
 * $\exists B \in \mathscr B : A \subseteq B$

Proof
Consider the ordered set $\struct {\mathscr I, \subseteq}$.

From Element of Finite Ordered Set is Between Maximal and Minimal Elements:
 * $\exists B \in \mathscr I : A \subseteq B$ and $B$ is maximal in $\struct {\mathscr I, \subseteq}$.

By definition of a base:
 * $B \in \mathscr B$