Dependent Subset Contains a Circuit

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

Let $\mathscr C$ denote the set of all circuits of $M$.

Let $A$ be a dependent subset.

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

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

From Element of Finite Ordered Set is Between Maximal and Minimal Elements:
 * $\exists C \in \mathscr I : C \subseteq A$ and $A$ is minimal in $\struct {\powerset S \setminus \mathscr I, \subseteq}$.

By definition of a circuit:
 * $C \in \mathscr C$