Can there be a model of the theory "$MK-Limitation$ $of$ $size+Subsets-Union$" having every proper class strictly smaller than the class $V$ of all sets being equinumerous to a set? provided of course that the model have at least one proper class that is strictly subnumerous to $V$.

Where $MK$ is $Morse$-$Kelley$ set theory, and $Subsets$ is the axiom asserting that every subclass of a set is a set.