Independent Subset Contains No Dependent Subset/Corollary 1

Theorem

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

Let $B \subseteq S$ be any base of $M$.

Then:

No dependent subset $D$ of $M$ is a subset of $B$.

Proof

By definition of matroid base:

$B$ is an independent subset of $M$

From Independent Subset Contains No Dependent Subset:

No dependent subset $D$ of $M$ is a subset of $B$.

$\blacksquare$