User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Axiom B4 Iff Axiom B5

Theorem
Let $S$ be a finite set.

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

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

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

Necessary Condition
Follows immediately from Axiom 4 and Axiom 5.

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