User:Leigh.Samphier/Sandbox/Proper Subset of Matroid Circuit is Independent

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

Let $C \subseteq S$ be a circuit of $M$.

Then:
 * every proper subset $A$ of $C$ is independent

Proof
By definition of a circuit of $M$:
 * $C$ is a minimum dependent subset of $M$

By definition of the minimum dependent subset of $M$:
 * every proper subset $A$ of $C$ is not a dependent subset

By definition of a dependent subset:
 * every proper subset $A$ of $C$ is an independent subset