Singleton is Dependent implies Rank is Zero

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

Let $x \in S$.

Let $\set x$ be dependent.

Then:
 * $\map \rho {\set x} = 0$

where $\rho$ denotes the rank function of $M$.

Proof
By definition of a dependent subset:
 * $\set x \notin \mathscr I$

Then: