To prove that a set exists, do I need to demonstrate that the set has a superset? As I read various examples of how to prove that sets with particular properties exist, I notice that a requirement seems to be that the set-to-be-constructed must be a subset of another set that I have already demonstrated to exist. Why must this be so?
To provide the most recent example I came across...I made a post asking about the formal statement for the construction of the cartesian product $S \times T$.
Rather than: $\exists C \forall z [z \in C \leftrightarrow \exists x \exists y (x \in S \land y \in T \land z=(x,y))]$, the correct answer is apparently:
$\exists C \forall z [z \in C \leftrightarrow \color{red}{z \in A} \land \exists x \exists y (x \in S \land y \in T \land z=(x,y))]$, where $A$ is a set that has been previously demonstrated to exist.
I interpreted this as saying "$C$ must be a subset of $A$ in order for $C$ to exist".
From the Axiom Schema of Separation/Comprehension, I know that every set has a subset. But the above example (and a few others I have come across) seems to say that every set must be the subset of another set. At first, I was tempted to say that "If all sets have subsets, then all sets are subsets", but I don't think that is a true implication.
What issues do we run into if we construct a set for which we cannot demonstrate has a superset? I assume that there must be some sort of contradiction that we risk encountering (perhaps related to Russell’s  paradox?) but I am uncertain. Any insight is greatly appreciated. Thanks!
 A: The condition of being a subset of some previously defined set is just important when you are trying to define a set using the axiom of separation. Naively, a set is a collection of all sets satisfying some given property, but we know from Russell's paradox that this is untenable. ZF fixes this issue by requiring that when we define a set by a comprehension that it is a subset of some other set.
But separation is not the only way to define new sets in ZF. And if we only had separation, we couldn't actually define any set, since there would be no supersets to start out with. So say we add an axiom that says the empty set exists. Then we're stuck again with just the empty set. But if we add some combination of pairing, union, power set and replacement, then we can define more and more sets. And then once we add infinity, we get an infinite set and it's off to the races.
Still, these construction axioms (at least if we don't include replacement), are pretty crude. They get us bigger and bigger sets, but don't let us slice and dice them logically... that's what separation is for. And so a large amount of definitions are ultimately definitions from separation, which is why you are seeing many definitions that require a stipulation that the set you are defining is a subset of some pre-defined set.
