Lets $\mathsf{MK^-}$ Be the theory "$\mathsf{MK}-\text{Limitation of size}+\text{Subsets}-\text{Union}$", removal of axiom of limitation of size and putting the axiom of subsets (the axiom asserting that every subclass of a set is a set) instead of it would make it possible for some classes to be equinumerous to sets and yet not being sets, since it is known that the axiom of subsets does not imply the assertion that every class that is equinumerous to a set is a set. So is it possible to have a model $M$ of $\mathsf{MK^-}$ such that we have all of the followings:

$M \vdash \exists P (\text {$P$ is a proper class} \wedge P < V)$
$M \vdash \forall P (\text{$P$ is a proper class} \wedge P < V \implies \exists x (x\in V \wedge \text{$P$ is equinumerous to x}))$

where "<" denotes "strict subnumerousity" defined in the customary manner; and $V$ is the class of all sets.

Also related to this is the following question:

If the above is possible then can we add the following:

$M \vdash \forall P (\text{$P$ is a proper class} \wedge P < V \implies \exists x (x=\{\{y\}| y \in P\} \wedge x \in V))$