Singleton is Independent implies Rank is One

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

Let $x \in S$.

Let $\set x$ be independent.

Then:
 * $\map \rho {\set x} = 1$

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

Proof
By definition of an independent subset:
 * $\set x \in \mathscr I$

Then: