Two questions about elements and subsets in ZFC Question 1: if $X$ is a set, are its elements sets? (in ZFC)
I believe the answer is yes. But where is the proof? Is it an axiom?
Question 2: If $X$ is a set, $Y$ is an entity of unknown nature which has elements, is the collection of elements of $X$ which are also elements of $Y$ a set?
If the answer is yes, then is it an axiom of ZFC? Can it be proved using the specification axiom?
Is it true in NBG? When $Y$ is a class?
Thank you all.
 A: Answer 1: The only thing that exists are sets; the axioms are (in some sense) describing what the rules we can do with sets to create more sets. More accurately, the axioms are about the things that exist.
Answer 2: You described the axiom of specification. This is indeed an axiom (schema) of ZFC.
A: Question 1: Yes, because in ZFC sets are the only kind of thing.
Question 2: Yes. This intersection is a subset of the set $X$, so it exists as a set by an Axiom of Specification.
For NBG, the answers are "Yes, because only sets can be elements of things" and "Yes, but now it's a corollary of the class existence theorem."
A: I agree that the answer to question 2 is "yes" (by what I would call the "separation" axiom). But I feel the other answers to question 1 are too glib.
Question 1 comes down to "does ZFC allow for the existence of things which aren't sets". It does not because of the axiom of extensionality. If $x$ is not a set then it has no elements, and therefore by extensionality it is the same as the empty set --- so it is a set after all. The axiom of extensionality tells us that everything is the same as a set.
