Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms/Condition 3 Implies Condition 4

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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:

\((C1)\)   $:$   \(\displaystyle \O \notin \mathscr C \)             
\((C2)\)   $:$     \(\displaystyle \forall C_1, C_2 \in \mathscr C:\) \(\displaystyle C_1 \neq C_2 \implies C_1 \not \subseteq C_2 \)             
\((C3'')\)   $:$     \(\displaystyle \forall X \subseteq S \land \forall x \in S:\) \(\displaystyle \paren{\forall C \in \mathscr C : C \not \subseteq X} \implies \paren{\exists \text{ at most one } C \in \mathscr C : C \subseteq X \cup \set x} \)             


Then:

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

Proof

$\blacksquare$

Sources