Let $\mathsf{MK^-}$ be the theory "$\mathsf{MK}-\text{Foundation}-\text{Limitation of size}-\text{Union}+\text{Subsets}$", where $\mathsf{MK}$ is Morse-Kelley set theory with axioms mentioned in:


<https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory>


Of course axioms of pairing and union and one direction of axiom of limitation of size are known to be redundant with this specific formulation of $\mathsf{MK}$.


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\in V (\text{$P$ is equinumerous to x}))$


where "<" denotes "strict subnumerousity" defined in the customary manner; and $V$ is the class of all sets. Given that, it is clear that $M$ cannot satisfy closure of replacement over sets. The general context of this question is about size of proper classes and to what extent that can be shared with size of sets in absence of Replacement and global choice.


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 \in V (x=\{\{y\}| y \in P\}))$


The special context of those questions raised when I was investigating an alternative to the axiom of limitation of size of $\mathsf{MK}$. A version that I've lately posted to FOM thread is referred to by the following link:


<http://www.cs.nyu.edu/pipermail/fom/2016-September/020073.html>


This version proves all axioms of Empty set, Pairing, Power, Separation (subsets), Replacement, Limitation of size, Global choice and Set union, so it is more powerful than the usual limitation of size axiom.


I've been trying to further simplify this axiom to the following:


$\forall x (x \in V \iff U(x) < V)$


However I couldn't prove Replacement nor union, so I was left with the above
situation (i.e., theory $\mathsf{MK^-}$); so if the possibilities that I've asked about specifically above were inconsistent then this would mean that this axiom can be simplified to the above version.



Proper classes subnumerous to $V$ in a model of a Morse-Kelley related theory
I believe the following gives a positive answer to your first question, modulo large cardinals:


Let $\kappa$ be measurable, and consider a [Prikry-generic](http://blog.assafrinot.com/?p=2156) extension of the universe $V[G]$. Look at $\mathcal{M}=(V\_{\kappa+1})^{V[G]}$. We can view this as a model of $MK^-$ by taking the proper classes to be exactly those elements of $\mathcal{M}$ of rank $\kappa$. In particular, the generic $G$ - which is a cofinal map from $\omega$ to $\kappa$ - is a "proper class" in the sense of $\mathcal{M}$, yet equinumerous with the $\mathcal{M}$-set $\omega$.


I strongly suspect that the measurable is unnecessary here - I'm just too lazy to think up the details for a ZFC-example (maybe something closer to Namba forcing?). The reason I'm using Prikry here is because Prikry forcing doesn't add any bounded-rank subsets of $\kappa$, which gives a fast (but probably unnecessary) proof that $\mathcal{M}$ is in fact a model of $MK^-$.


