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 $S$.

Necessary Condition
Follows immediately from Axiom $(\text B 4)$ and Axiom $(\text B 5)$.

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

It follows that $\mathscr B$ satisfies the base axiom: