Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms/Condition 2 Implies Condition 3

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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:

\((C1)\)   $:$   \(\ds \O \notin \mathscr C \)             
\((C2)\)   $:$     \(\ds \forall C_1, C_2 \in \mathscr C:\) \(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \)             
\((C3')\)   $:$     \(\ds \forall C_1, C_2 \in \mathscr C:\) \(\ds C_1 \ne C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : y \in C_3 \subseteq \paren {C_1 \cup C_2} \setminus \set z \)             


Then:

$\mathscr C$ satisfies the circuit axioms:
\((C1)\)   $:$   \(\ds \O \notin \mathscr C \)             
\((C2)\)   $:$     \(\ds \forall C_1, C_2 \in \mathscr C:\) \(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \)             
\((C3'')\)   $:$     \(\ds \forall X \subseteq S \land \forall x \in S:\) \(\ds \paren {\forall C \in \mathscr C : C \nsubseteq X} \implies \paren {\exists \text{ at most one } C \in \mathscr C : C \subseteq X \cup \set x} \)             


Proof

$\blacksquare$

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