Minimal size of a proper class Assume ZFC consistent. Let $\mathcal c$ be some infinite cardinality and $\mathfrak F$ a family of models of ZFC of cardinality $\mathcal c$.
Let $\mathcal S = \left\{ \left\{ \{ x | x \in_{\mathcal M} C \} | C \textbf{ is proper class in } \mathcal M \right\} | \mathcal M \in \mathfrak F \right\}$
So $\mathcal S$ is a collection of sets that are a proper class of some ZFC model with cardinality $\mathcal c$.
Can anything be said about the minimal cardinality of the elements of $\mathcal{S}$? Is it necessarily $\mathcal c$ or can it be smaller?
 A: As Taneli Huuskonen says, if $M$ is a model of ZFC then (externally) there is a bijection between $M$ and $Ord^M$. The proof is as follows.
Since $M$ is a model of ZFC, $M$ thinks that every set in $M$ is well-orderable. This means that for every $m\in M$, the set $\{a\in M: a\in^M m\}$ injects into $Ord^M$.
Now consider the levels, in the sense of $M$, of the cumulative hierarchy; that is, the sets $(V_\alpha)^M$ for $\alpha\in Ord^M$. There are $\vert Ord^M\vert$-many of these, and each one has (external) cardinality $\le\vert Ord^M\vert$; so their union also has cardinality $\le \vert Ord^M\vert$ since $Ord^M$ is infinite. But this means that $\vert M\vert\le \vert Ord^M\vert$, and since $Ord^M\subseteq M$ this gives us $\vert M\vert=\vert Ord^M\vert$.

Now, some caveats with the same theme - that definability is key. 
First of all, the class in question has to be definable in order for this to make any sense. If $X\subseteq M$ is not definable in $M$ (that is, not a class in $M$ in the usual sense of the word) then there is very little we can conclude about the cardinality of $X$ versus that of $M$. For example, we can have an uncountable model $M$ of ZFC where $Ord^M$ has (external) cofinality $\omega$; then there is a countable set of $M$-ordinals which has elements of arbitrarily large $M$-rank. Of course, such a set will not be definable in $M$.
Second, things get interesting once we talk about definable bijections/injections. The argument above shows that $\vert M\vert=\vert Ord^M\vert$ whenever $M$ is a model of ZFC. But we can also ask:


*

*If $M$ is a model of ZFC, is there necessarily a definable-in-$M$ class bijection between $Ord^M$ and $M$? Exercise: it's enough to have a definable-in-$M$ injection from $M$ to $Ord^M$ - think about the Mostowski collapse ...

*If $M$ is a model of ZFC, is there necessarily a definable-in-$M$ surjection from $Ord^M$ to $M$? This would clearly be implied by a positive answer to the previous question.
It turns out that the answer to even the second question may be "no." So once we restrict to definable comparisons, things can get much more interesting.
A: It is provable in ZFC that there is a definable surjection from any proper class $C$ onto the ordinals (collapse the class of ranks of the elements of $C$). OTOH, the cardinality of the ordinals of a model of ZFC is the same as the cardinality of the whole model. Thus, every proper class of a set model has the same cardinality as the model itself.
