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:

$B \in \mathscr B$

$\blacksquare$