Singleton is Independent implies Rank is One/Corollary

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

Let $x \in S$.

Then:
 * $\set x$ is an independent subset $\map \rho {\set x} = 1$

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

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

From Singleton is Independent implies Rank is One:
 * if $\set x \in \mathscr I$ then $\map \rho {\set x} = 1$

From Singleton is Dependent implies Rank is Zero:
 * if $\set x \notin \mathscr I$ then $\map \rho {\set x} = 0$

It follows that:
 * $\set x \in \mathscr I$ $\map \rho {\set x} = 1$