Let $M$ be a transitive proper class satisfying ZF. Prove that \begin{equation} \forall x \subset M\left(\exists y\in M\left(x\subseteq y \right)\right) \ \ \ \ (*) \end{equation} My progress
I can prove that if $M$ is transitive, $M$ satisfies $(*)$ and Comprehension Scheme, then $M$ satisfies ZF. Particularly I used $(*)$ to prove $M$ satisfies Replacement, Power, Pairing, Union and Infinity. Thus I think I must use all of them to prove the inverse, but unclear how.
The key idea, in my mind, is to "overshoot": don't try to get a set which is "close to $x$" in any sense, but rather look for something so big that everything in $x$ is guaranteed to enter it.
I don't want to fully give away the problem, but here's a hint: if $x$ is a set and $M$ is a proper class, then the elements of $x$ have bounded rank in the sense of $M$. Can you think of a set in $M$ which contains all the elements of $M$ with rank some fixed ordinal (or rather, with rank at most some fixed ordinal)?
Note: The statement is of course false if we replace "$\subset$" by "$\in$" - e.g. we may not even have $\mathcal{P}(\omega)=\mathcal{P}(\omega)^M$, but clearly any $x\in\mathcal{P}(\omega)$ is a subset of $M$.