User:Leigh.Samphier/Matroids/Equivalence of Definitions of Matroid Base Axioms/Axiom B1 Iff Axiom B4

Theorem
Let $S$ be a finite set.

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

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

From Set of Matroid Bases iff Axiom $( \text B 1)$:
 * there exists a matroid $M = \struct{S, \mathscr I}$ such that $\mathscr B$ is the set of bases of $M$.

Let $B_1, B_2 \in \mathscr B$.

Let $x \in B_1 \setminus B_2$.

From Matroid Base Union External Element has Fundamental Circuit:
 * there exists a fundamental circuit $\map C {x, B_2}$ of $M$ such that $x \in \map C {x, B_2} \subseteq B_2 \cup \set x$

By definition of set intersection:
 * $x \in B_1 \cap \map C {x, B_2}$

From Element of Matroid Base and Circuit has Substitute:
 * $\exists y \in \map C {x, B_2} \setminus B_1 : \paren{B_1 \setminus \set x} \cup \set y \in \mathscr B$

We have:

From Matroid Base Substitution From Fundamental Circuit:
 * $\paren{B_2 \setminus \set y} \cup \set x \in \mathscr B$

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

Sufficient Condition
Follows immediately from Axiom $(\text B 4)$ and Axiom $(\text B 1)$.