Equivalence of Definitions of Matroid Rank Axioms/Lemma 1

Theorem
Let $S$ be a finite set.

Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.

Let $\rho$ satisfy the rank axioms:

Then:
 * $\forall A, B \subseteq S: A \subseteq B \implies \map \rho A \le \map \rho B$