Is an ordered pair a set? I'm having a problem with a homework. The task is to show whether the expression
$$\bigcup\mathcal{A}\times\bigcap\mathcal{B}\subseteq\bigcup\{a\times b\mid a\in\mathcal{A}\land b\in\mathcal{B}\}$$
is true for all families $\mathcal{A}$ and $\mathcal{B}$. In order for operation standing in front of the right set to make sense, the set $\{a \times b \mid a \in \mathcal{A} \land b \in \mathcal{B}\}$ must be a family of sets. But in order for that to make sense "$a \times b$" must be considered a set as well. So the question is - is "$a \times b$" which is an ordered pair can be considered a set? If that is the case, how?
 A: Kuratowski's definition for ordered pair is the following:
$$(a, \ b) \; := \  \{ \{ a \}, \ \{ a, \ b \} \}.$$
With this definition an ordered pair is a set. Actually every object is a set in a mathematics which based on set theory.
$A\times B$ is the Cartesian product of two sets. By definition it means that
$$A\times B := \{\,(a,b)\mid a\in A \ \mbox{ and } \ b\in B\,\}.$$
Using ordered pair definition you can write $A \times B$ into the form
$$A\times B = \{\, \{ \{ a \}, \ \{ a, \ b \} \} \mid a\in A \ \mbox{ and } \ b\in B\,\}.$$
$A \times B$ exists, because it's a subset of $\wp\left(\wp(A \cup B)\right)$, since if $a\in A$, $b\in B$ then $\{a\},\{a,b\} \in \wp(A \cup B)$ and therefore $(a,b) = \{ \{ a \}, \ \{ a, \ b \} \} \in \wp\left(\wp(A \cup B)\right)$. From here the axiom of power set and the axiom of union provides the existence of Cartesian product. At last because the Cartesian product of two sets is the set of all ordered pairs from those two sets, therefore we can talk about ordered pairs. Of course we need axiom of empty set and axiom of pairing, if we want to construate ordered pairs. 
