Intersection of Zermelo-Fraenkel universes containing all ordinals I am reading my first Set Theory book (Set Theory and the Continuum Problem, Smullyan, 2010) and I find the subject pretty interesting. I am not a professional mathematician and can only study maths in my spare time.
After presenting ordinal numbers and transfinite recursion, the author presents the class $R_\Omega$ of all sets with rank. $R_\Omega$ is defined as the union of a sequence of sets $R_\alpha$, $\alpha$ being an ordinal number. The $R_\alpha$'s are constructed recursively by taking the power set of the previous set in the sequence or, in the case of limit ordinal, the union of all previous sets. A set has rank $\alpha$ iff it lies in $R_{\alpha+1}-R_\alpha$.
As an exercise the author asks the reader to show that $R_\Omega$ is the intersection of all subclasses $C$ of $V$ ($V$ being The universal class, the class of all sets) such that $C$ is a Zermelo-Fraenkel (ZF) universe containing all ordinals of $V$.
Previously the author has shown that $R_\Omega$ is in fact a ZF universe and since each ordinal has rank itself it is clear that $R_\Omega$ includes the intersection. However it is not clear to me why the converse is true. Why should $R_\Omega$ be the smallest ZF class containing all ordinals?
Now I am new to the subject and doing a little bit of research I can see that what the author calls $R_\Omega$ is often called V in literature and can be approached by different ways that I do not understand at this point. But I hope this question makes sense, any help appreciated.
 A: I glimpsed your textbook (but an older version. 2010 version is not available to me) and I found that Smullyan and Fitting distinguish ZF-universes and first-order ZF-universes. They stated separation axiom and replacement axiom as a second-order form, and differentiate these axioms with its first-order counterparts.
(Note that some textbooks (e.g., Jech) also use this kind of description for first-order Separation Replacement, although most of them give detail on its first-order nature.)
That is, Smullyan and Fitting assume second-order ZF. ZF-models in your textbook are closed under an arbitrary subset of their elements (or swelled under their terminology.) Hence they are closed under the true powersets, as I and Asaf mentioned in the comment.
Then the proof of your problem is direct: If $W$ is a ZF-model which contains every ordinal, then $R_\alpha\subseteq W$ for every $\alpha$ by induction on $\alpha$. Hence $V\subseteq W$.
A: Here is my solution to my own question, based on the comments I got.
I want to show that $R_\Omega$ is in the intersection of all subclasses $C$ of $V$ which are ZF universes and contain all ordinals. To show this it suffices to show that $R_\alpha \subseteq C$ for all the considered $C$'s and all ordinals $\alpha$, since $R_\Omega$ is the union of the $R_\alpha$'s.
Consider a ZF subclass $C$ of $V$ containing all ordinals. First we show that $R_\alpha \in C$ by transfinite induction on $\alpha$.

*

*$R_0 = 0 \in C$ by the Empty Set Axiom.

*If $R_\alpha \in C$ then $R_{\alpha+1} = P(R_\alpha) \in C$ by Power Axiom.

*Now the limit case. Suppose that for every $\beta < \alpha$ we have $R_\beta \in C$. Define the function $F$ from $\alpha$ to $C$ such that $F(\beta)=R_\beta$, for all ordinals $\beta \in \alpha$. Since $C$ contains all ordinals it contains $\alpha$ so $\alpha$ is a set and we can use the Substitution Axiom to conclude that $F''(\alpha):=\{F(\beta) | \beta < \alpha\}$ is a set. Then by Union Axiom $\cup F''(\alpha) = R_\alpha$ is a set, i.e. $R_\alpha \in C$ which concludes the induction.

Now for every ordinal $\alpha$ we have $R_\alpha \subseteq R_{\alpha+1}$ and $R_{\alpha+1} \in C$. So by the Swelled Axiom, $R_\alpha \subseteq C$, which is what we wanted.
Please let me know if you spot mistakes.
