\begin{definition}[Definition:Base Axiom (Matroid)]
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''' {{iff}}:
{{begin-axiom}}
{{axiom | n = \text B 1
        | q = \forall B_1, B_2 \in \mathscr B
        | mr= x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \cup \set y} \setminus \set x \in \mathscr B
}}
{{end-axiom}}
\end{definition}