Equivalence of Definitions of Matroid/Definition 1 implies Definition 2

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

Let $M$ also satisfy:

Then $M$ satisfies:

Proof
Since:
 * $\forall U, V \in \mathscr I : \size U = \size V + 1 \implies \size V < \size U$

If follows that if $M$ satisfies condition $(\text I 3)$ then $M$ satisfies condition $(\text I 3')$.