Equivalence of Definitions of Matroid Circuit Axioms

Theorem
Let $S$ be a finite set.

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

Condition 1
$\mathscr C$ satisfies the circuit axioms:

Condition 2
$\mathscr C$ satisfies the circuit axioms:

Condition 3
$\mathscr C$ satisfies the circuit axioms:

Condition 4
$\mathscr C$ is the set of circuits of a matroid on $S$