User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms

Theorem
Let $S$ be a finite set.

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

Condition 1
$\mathscr B$ satisfies any one of the base axioms:

Condition 2
$\mathscr B$ satisfies all of the base axioms.

Condition 3
$\mathscr B$ is the set of bases for some matroid on $S$.

Axiom $(\text B 1)$ iff Axiom $(\text B 2)$
Axiom $(\text B 1)$ holds Axiom $(\text B 2)$ holds follows immediately from the lemma 6.

Axiom $(\text B 5)$ iff Axiom $(\text B 6)$
Axiom $(\text B 5)$ holds Axiom $(\text B 6)$ holds follows immediately from the lemma 6.