Studying Set Theory by Karel Hrbacek, I started to question the existence of a set S that contains exactly n given sets, a set of the form:
$$\forall X_{1}, X_{2}, \cdots ,X_{n} \space \exists S \space\forall a (a \in S \iff a = X_{1} \space \vee a=X_{2} \space \vee \cdots \vee \space a= X_{n} )$$
I felt this could be done using the Axioms of Pair and Union, so i tried this:
$$\forall X_{1} \space \exists Y_{1} \space \forall a \space(a \in Y_{1} \iff a = X_{1})$$ $$\forall X_{2} \space \exists Y_{2} \space \forall a \space(a \in Y_{2} \iff a = X_{2})$$ $$\vdots $$ $$\forall X_{n} \space \exists Y_{n} \space \forall a \space(a \in Y_{n} \iff a = X_{n})$$
Now, it is necessary to construct a set $Z$ that contains $Y_{1}, Y_{2}, \cdots, Y_{n} $ and apply the operation of union to $Z$, so there exists a set $ \cup Z = S$ that contains exactly the wanted elements. I think that the problem here is in fact the construction of $Z$, but this is redundant because the construction of $Z$ is a problem equivalent to the construction of $S$.
I don't know if my actual knowledge is enough to answer this question, so any help is appreciated.