User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Axiom B1 Iff Axiom B3

Theorem
Let $S$ be a finite set.

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

Axiom 1
$\mathscr B$ is said to satisfy the base axiom :

Axiom 3
$\mathscr B$ is said to satisfy the base axiom :

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

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

Sufficient Condition
By choosing $y = \map \pi x$ in Axiom 3, Axiom 1 follows immediately.