  We solve the satisfiability problem for a three-sorted fragment of set theory
(denoted $3LQST_0^R$), which admits a restricted form of quantification over
individual and set variables and the finite enumeration operator $\{\text{-},
\text{-}, \ldots, \text{-}\}$ over individual variables, by showing that it
enjoys a small model property, i.e., any satisfiable formula $\psi$ of
$3LQST_0^R$ has a finite model whose size depends solely on the length of
$\psi$ itself. Several set-theoretic constructs are expressible by
$3LQST_0^R$-formulae, such as some variants of the power set operator and the
unordered Cartesian product. In particular, concerning the unordered Cartesian
product, we show that when finite enumerations are used to represent the
construct, the resulting formula is exponentially shorter than the one that can
be constructed without resorting to such terms.
