User:Leigh.Samphier/Matroids/Matroid Bases Iff Satisfies Formulation 5 of Matroid Base Axiom/Necessary Condition

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Theorem

Let $M = \struct {S, \mathscr I}$ be a matroid.

Let $\mathscr B$ be the set of bases of the matroid $M$.

Then $\mathscr B$ satisfies formulation $5$ of base axiom:

\((\text B 5)\)

$:$

\(\ds \forall B_1, B_2 \in \mathscr B:\)

\(\ds x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_2 \setminus \set y} \cup \set x \in \mathscr B \)

Proof

$\blacksquare$