Element is Loop iff Member of Closure of Empty Set

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

Let $x \in S$.

Then:
 * $x$ is a loop $x \in \map \sigma \O$

where $\map \sigma \O$ denotes the closure of the empty set.

Proof
From Element is Loop iff Rank is Zero:
 * $x$ is a loop $\map \rho {\set x} = 0$

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

Now:

The result follows.