User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Set of Matroid Bases Iff Axiom B1

Theorem
Let $S$ be a finite set.

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

Then $\mathscr B$ is the set of bases of a matroid on $S$ $\mathscr B$ satisfies the base axiom:

Necessary Condition
Let $\mathscr B$ be the set of bases of the matroid on $M = \struct{S, \mathscr I}$

Sufficient Condition
Let $\mathscr B$ satisfies the base axiom: