# Leigh.Samphier/Sandbox/Definition:Base Axiom (Matroid)/Definition 7

Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$.
$\mathscr B$ is said to satisfy the base axiom if and only if:
 $(\text B 7)$ $:$ $\displaystyle \forall B_1, B_2 \in \mathscr B:$ $\displaystyle \exists \text{ a bijection } \pi : B_1 \setminus B_2 \to B_2 \setminus B_1 : \forall x \in B_1 \setminus B_2 : \paren {B_2 \setminus \set {\map \pi x} } \cup \set x \in \mathscr B$