User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Circuit Axioms/Formulation 2 Implies Formulation 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 (formulation 2):

Then:
 * $\mathscr C$ satisfies the circuit axioms (formulation 1):

Proof
Let $\mathscr C$ satisfy the circuit axioms $(\text C 1)$, $(\text C 2)$ and $(\text C 3')$.

We need to show that $\mathscr C$ satisfies circuit axiom: