User:Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 1 Iff Definition 3

Theorem
Let $S$ be a finite set.

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

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

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

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

$\dots$

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

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