Rank of Empty Set is Zero

Theorem
Let $M = \struct {S, \mathscr I}$ be a matroid.

Let $\rho : \powerset S \to \Z$ be the rank function of $M$.

Then:
 * $\map \rho \O = 0$

Proof
By matroid axiom $(\text I 1)$:
 * $\O$ is independent

From Rank of Independent Subset Equals Cardinality:
 * $\map \rho \O = \size \O$

From Cardinality of Empty Set:
 * $\size \O = 0$

The result follows.