Equivalence of Definitions of Matroid/Definition 4 implies Definition 1

Theorem
Let $M = \struct {S, \mathscr I}$ be an independence system.

Let $M$ also satisfy:

Then $M$ satisfies:

Proof
Let $M$ satisfy condition $(\text I 3''')$.

Let $U, V \in \mathscr I$ such that $\size V < \size U$.

Let $W$ be a maximal independent subset of $U \cup V$ containing $U$.

Then:
 * $\size U \le \size W$

By condition $(\text I 3''')$:
 * $V$ is not a maximal independent subset of $U \cup V$

Then:
 * $\exists x \in \paren {U \cup V} \setminus V$ such that $V \cup \set x \in \mathscr I$

From Set Difference with Union is Set Difference:
 * $\exists x \in U \setminus V$ such that $V \cup \set x \in \mathscr I$

It follows that $M$ satisifies $(\text I 3)$