Equivalence of Definitions of Matroid/Definition 3 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 $\size V < \size U$.

By condition $(\text I 3'')$:
 * $\exists Z : \exists Z \subseteq U \setminus V : \paren{V \cup Z \in \mathscr I} \land \paren{ \size {V \cup Z} = \size U}$

Then:
 * $V \cup Z \neq V$

From Union with Empty Set:
 * $Z \neq \O$

Then:
 * $\exists x : x \in Z$

It follows that:
 * $V \cup \set x \subseteq V \cup Z$

From independence system axiom $(\text I 2)$:
 * $V \cup \set x \in \mathscr I$

By definition of a subset:
 * $x \in U \setminus V$

It follows that $M$ satisfies condition $(\text I 3)$.