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:

Let $C_1, C_2 \in \mathscr C : C_1 \ne C_2$.

Let $z \in C_1 \cap C_2$.

From circuit axiom $(\text C 2)$:
 * $C_2 \nsubseteq C_1$

By definition of subset and set difference:
 * $\exists w \in C_2 \setminus C_1$

From circuit axiom $(\text C 3')$:
 * $\exists C_3 \in \mathscr C : w \in C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z$

It follows that $\mathscr C$ satisfies circuit axiom $(\text C 3)$