Singleton is Dependent implies Rank is Zero/Corollary

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

Let $x \in S$.

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

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

Proof
By definition of a loop:
 * $x$ is a loop $\set x \notin \mathscr I$

By definition of the rank function:
 * $\map \rho {\set x} = \max \set{\size A : A \in \powerset{\set x} \land A \in \mathscr I}$

From Power Set of Singleton:
 * $\powerset {\set x} = \set{\O, \set x}$

Let $\set x \notin \mathscr I$

Then:

Let $\set x \in \mathscr I$

Then:

It follows that:
 * $\set x \notin \mathscr I$ $\map \rho {\set x} = 0$