Terminology:
Let $\mathfrak{A} = (A,\in^\mathfrak{A})$ be a model of ZFC (for simplicity let it be an "inner model", i.e. it is itself an element in the universe $\mathfrak{U}$).
A unary predicate over $A$ I call a "class", i.e. the classes are the subsets $K \subseteq A$ s.t. $K = [\![\varphi(x)]\!]^\mathfrak{A}$ for some first-order expression $\varphi \in L^{\{\in\}}$ having no free variables other than $x$, where $[\![\varphi(x)]\!]^\mathfrak{A} := \{a \in A \mid \mathfrak{A} \models \varphi[a]\}$.
A finite class is just a class $K$ that is finite. For simplicity, we can forget about the general case of a finite cardinal and restrict ourselves to the case of $|K|$ being a "counting number" n $\geq$ 2, maybe even just the case $|K| =$ 2, as the answer to that should be enough for the general case I believe.
A set $a \in A$ s.t. $\{a\}$ is a class I call a "definable set", i.e. it is a set that can be "pinned down" using some first order expression.