Cycle Matroid is Matroid

Theorem
Let $G = \struct {V, E}$ be a graph.

Let $\struct {E, \mathscr I}$ be the cycle matroid of $G$.

Then $\struct {E, \mathscr I}$ is a matroid.

Proof
It needs to be shown that $\mathscr I$ satisfies the matroid axioms $(\text I 1)$, $(\text I 2)$ and $(\text I 3)$.

Matroid Axiom $(\text I 1)$
$\ds \O \subset E$ is the empty subgraph, which does not contain any cycles.

Hence, by definition of the cycle matroid, $\ds \O \in \mathscr I$.