Equivalence of Definitions of Matroid Circuit Axioms/Condition 3 Implies Condition 1

Theorem
Let $S$ be a finite set.

Let $\mathscr C$ be a non-empty set of subsets of $S$.

Let $\mathscr C$ satisfy the circuit axioms:

Then:
 * $\mathscr C$ is the set of circuits of a matroid on $S$