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

Theorem
Let $S$ be a finite set.

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

Axiom 1 iff Axiom 2
Axiom 1 holds Axiom 2 holds follows immediately from the lemma.

Axiom 5 iff Axiom 6
Axiom 5 holds Axiom 6 holds follows immediately from the lemma.