The Class Of All Possible Worlds Is the class of all possible worlds a proper class? I guess that you could say that under a function any possible world can be defined as a set, and hence the class of all possible worlds would be bijective to the universe of all sets which is a proper class.
Like the following.
Assume, that for all possible worlds $W$ there exists a bijective function $F$ such that it turns a possible world $W$ to a corresponding set $X$.
$$\forall W[\;\exists F: W\to X\;]$$
Now assume there exists a class of all possible worlds $W$ called $\mathscr{L}$, well if that is true then one could also say the exists a class of all sets $V$ (Universe of all sets). Well given the statement above would it also be reasonable to say that there exists a bijective function $G: \mathscr{L}\to  V$, and hence couldn't the class of all possible worlds $\mathscr{L}$ be classified as a proper class because $V$ is also a proper class?
But is this proof correct?
 A: There is no such thing as "possible worlds," therefore there is no "class of all possible worlds." That's a semantical addition. In theory, what we have is either referential Propositional Logic (+ Transitivity Predicate), or its first-order counterpart.
What you are referring to as "Possible world" is a Kripkean coinage (The so-called Kripke semantics) to make sense of ridiculously nested-valuations. What you can really ask is: is "the class of all nested valuation" a proper class... I haven't got a clue what that would even mean.
By the way, I am not saying semantics is useless, etc. What I am saying is: your question is equivalent to asking, "what's the weight of justice." Just like Justice has no weight because it's an abstract concept. There is no class of "Possible worlds" because it's an abstract concept we use to make sense of the apparently unintuitive syntax of modal logic.
For example, when talking about Modal Propositional logic the So-called set of possible worlds $W$ in the tuple $\langle W, R, \bar a, v \rangle$ is just some finite subset of $\mathbb N$. From this we get reference dependent valuation in the ordered pair $(v(P),i)$; where $i\in W \subseteq \mathbb N$. Now, if you mean a proper subclass in this sense of "Possible world" i.e. $W\subseteq\mathbb N$ that I just explained, then that becomes trivial since we can forget about possible worlds, and just consider some arbitrary subset of $\mathbb N$
One trivial fact comes to mind, the class [or in terms of ZFC, set] of set "All possible worlds" and $\scr P(\mathbb N)$ are equidimensional. One more, perhaps, even more trivial fact is that the two are identical. That is, the "class of all possible world" if it were to exist would just be $\scr P(\mathbb N)$
As for the last part, is there a bijection between the class of all possible worlds and the class of all sets? That's resounding no, since $|\scr P(\mathbb N)|\lt|\scr P(\mathbb R)|$
