Independent Subset Contains No Dependent Subset/Corollary 2

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Theorem

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

Let $X \subseteq S$ be any independent subset of $M$.

Then:

No circuit $C$ of $M$ is a subset of $X$.

Proof

Let $C$ be a circuit of $M$.

By definition of matroid circuit:

$C$ is a dependent subset of $M$

From Independent Subset Contains No Dependent Subset:

$C$ is not a subset of $X$.

The result follows.

$\blacksquare$