Equivalence of Definitions of Matroid Circuit Axioms/Condition 4 Implies Condition 1

Theorem
Let $S$ be a finite set.

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

Let $\mathscr C$ be the set of circuits of a matroid $M = \struct{S, \mathscr I}$ on $S$

Then:
 * $\mathscr C$ satisfies the circuit axioms:

Proof
Let $\rho$ denote the rank function of the matroid $M$.

$C_1, C_2 \in \mathscr C:$
 * $C_1 \neq C_2$

and
 * $\exists z \in C_1 \cap C_2 : \nexists C_3 \in \mathscr C : C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z$

From Dependent Subset Contains a Circuit:
 * $\paren{C_1 \cup C_2} \setminus \set z$ is an independent subset.

We have: