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 $S$.

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

Definition 1 implies Definition 3
Let $\mathscr B$ satisfy the base axiom:

Definition 3 implies Definition 1
Follows immediately from Definition 3 and Definition 1.

Definition 1 implies Definition 4
Let $\mathscr B$ satisfy the base axiom:

Definition 4 implies Definition 5
Follows immediately from Definition 4 and Definition 5.

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

Definition 5 implies Definition 7
Let $\mathscr B$ satisfy the base axiom:

Definition 7 implies Definition 1
Let $\mathscr B$ satisfy the base axiom: