User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Lemma 2

Theorem
Let $S$ be a finite set.

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

Let $\mathscr B$ satisfy the base axiom:

Then:
 * $\forall B_1, B_2 \in \mathscr B : \card{B_1} = \card{B_2}$

where $\card{B_1}$ and $\card{B_2}$ denote the cardinality of the sets $B_1$ and $B_2$ respectively.