[proofplan]
We prove both implications, using the stated convention that $\mathcal C$ is closed under matroid isomorphism. If $M$ lies in the minor-closed class $\mathcal C$, then every minor of $M$ also lies in $\mathcal C$, so a minor isomorphic to an excluded minor cannot occur. Conversely, if $M$ does not lie in $\mathcal C$, then the finite set of minors of $M$ contains at least one matroid outside $\mathcal C$; choosing one with minimum ground-set size forces all of its proper minors to lie in $\mathcal C$, so it is an excluded minor.
[/proofplan]
[step:Rule out excluded minors when $M$ belongs to the class]
Assume $M \in \mathcal C$. Let $N$ be a minor of $M$. Since $\mathcal C$ is closed under taking minors, $N \in \mathcal C$.
Suppose, for contradiction, that $N$ is isomorphic to some matroid $E \in \mathcal E$. Since $\mathcal C$ is closed under matroid isomorphism by hypothesis and $N \in \mathcal C$, the isomorphism $N \cong E$ implies $E \in \mathcal C$. This contradicts the defining condition $E \notin \mathcal C$ for excluded minors. Therefore no minor of $M$ is isomorphic to a matroid in $\mathcal E$.
[/step]
[step:Choose a minimal forbidden minor when $M$ is outside the class]
Assume $M \notin \mathcal C$. For any finite matroid $Q$, let $E(Q)$ denote the ground set of $Q$. Define
\begin{align*}
\mathcal F := \{N : N \text{ is a minor of } M \text{ and } N \notin \mathcal C\}.
\end{align*}
The class $\mathcal F$ is nonempty because $M$ is a minor of itself and $M \notin \mathcal C$.
Since $M$ is finite, every minor of $M$ is determined by a pair of disjoint subsets of $E(M)$, one subset to delete and one subset to contract. There are only finitely many such pairs, so $M$ has only finitely many minors. Hence there exists $N_0 \in \mathcal F$ such that the ground-set cardinality $|E(N_0)|$ is minimal among all members of $\mathcal F$.
[guided]
We now isolate the smallest obstruction inside $M$. For any finite matroid $Q$, write $E(Q)$ for its ground set. Define
\begin{align*}
\mathcal F := \{N : N \text{ is a minor of } M \text{ and } N \notin \mathcal C\}.
\end{align*}
This is the collection of minors of $M$ that still fail to belong to the class $\mathcal C$.
The set $\mathcal F$ is nonempty: the matroid $M$ is a minor of itself, obtained by performing no deletions and no contractions, and by assumption $M \notin \mathcal C$. Because $M$ is finite, its ground set $E(M)$ has finitely many subsets. Each minor of $M$ is obtained by choosing two disjoint subsets of $E(M)$, deleting one of them, and contracting the other. Thus there are only finitely many possible minors before passing to isomorphism.
Therefore the finite nonempty collection $\mathcal F$ contains an element with minimum ground-set size. Choose such a matroid and denote it by $N_0$. Thus $N_0$ is a minor of $M$, $N_0 \notin \mathcal C$, and for every $N \in \mathcal F$ we have
\begin{align*}
|E(N_0)| \leq |E(N)|.
\end{align*}
The point of minimizing $|E(N_0)|$ is that any proper minor of $N_0$ has strictly smaller ground set, so it cannot still lie outside $\mathcal C$.
[/guided]
[/step]
[step:Show that the minimal forbidden minor is excluded]
We claim that every proper minor of $N_0$ belongs to $\mathcal C$. Let $P$ be a proper minor of $N_0$. Since $N_0$ is a minor of $M$ and taking minors is transitive, $P$ is a minor of $M$. Also, because $P$ is a proper minor of $N_0$, its ground set satisfies
\begin{align*}
|E(P)| < |E(N_0)|.
\end{align*}
If $P \notin \mathcal C$, then $P \in \mathcal F$, contradicting the minimality of $|E(N_0)|$. Hence $P \in \mathcal C$.
We have shown that $N_0 \notin \mathcal C$ and every proper minor of $N_0$ belongs to $\mathcal C$. By the definition of $\mathcal E$, this means $N_0 \in \mathcal E$.
[/step]
[step:Conclude the equivalence]
Since $N_0$ is a minor of $M$ and $N_0 \in \mathcal E$, the matroid $M$ has a minor isomorphic to a matroid in $\mathcal E$, namely $N_0$ itself. Thus, if no minor of $M$ is isomorphic to a matroid in $\mathcal E$, then $M \in \mathcal C$.
Together with the first implication, this proves that $M \in \mathcal C$ if and only if no minor of $M$ is isomorphic to a matroid in $\mathcal E$.
[/step]
