User:Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5

Theorem
Let $S$ be a finite set.

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

Definition 4
$\mathscr B$ is said to satisfy the base axiom :

Definition 5
$\mathscr B$ is said to satisfy the base axiom :

Necessary Condition
Follows immediately from Definition 4 and Definition 5.

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