Equivalence of Definitions of Matroid Rank Axioms/Condition 3 Implies Condition 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.

Condition 1
$\rho$ satisfies the rank axioms:

Condition 2
$\rho$ satisfies the rank axioms:

Condition 3
$\rho$ is the rank function of a matroid on $S$