Matroid Contains No Loops iff Empty Set is Flat

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

Then:
 * $M$ contains no loops the empty set is flat.

Proof
Let $\rho: \powerset S \to \Z$ denote the rank function of $M$.

Necessary Condition
Let $M$ contain no loops.

By definition of a loop:
 * $\forall x \in S : \set x \in \mathscr I$

Let $x \in S \setminus \O$.

Then:

It follows that $\O$ is a flat subset by definition.

Sufficient Condition
Let $\O$ be a flat subset.

Let $x \in S$.

From Set Difference with Empty Set is Self
 * $x \in S \setminus \O$

From Element is Loop iff Rank is Zero:
 * $x$ is not a loop

It follows that $M$ contains no loops.