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 the base axiom $(\text B 1)$

Condition 2
$\mathscr B$ satisfies the base axiom $(\text B 2)$

Condition 3
$\mathscr B$ satisfies the base axiom $(\text B 3)$

Condition 4
$\mathscr B$ satisfies the base axiom $(\text B 4)$

Condition 5
$\mathscr B$ satisfies the base axiom $(\text B 5)$

Condition 6
$\mathscr B$ satisfies the base axiom $(\text B 6)$

Condition 7
$\mathscr B$ satisfies the base axiom $(\text B 7)$

Condition 8
$\mathscr B$ is the set of bases for some matroid on $S$.

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

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