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Im reading Set theory of Jech and its said that

Let $X,Y$ sets, then $X\times Y:=\{(x,y):x\in X\land y\in Y\}$. Then $X\times Y$ is a set because $X\times Y\subseteq\wp(\wp(X\cup Y))$.

Note: here $\wp$ means "power set". It is also assumed that $(x,y):=\{\{x\},\{x,y\}\}$.

Im not sure why if something is contained in a set then it is also a set, I assume that it is because there exists a formula such that $X\times Y$ could be defined by the axiom schema of separation. Im right, or there is other reasons to say that $X\times Y$ is a set?

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You're right -- it's the axiom schema of separation that is at play here. In particular $$ X \times Y = \{ z \in \mathcal P (\mathcal P(X \cup Y)) \mid \exists x \in X \exists y \in Y \colon z = (x,y) \}. $$ Here "$z= (x,y)$" is an abbreviation for the obvious (but lenghty) $\Delta_0$-formula $\phi(z,x,y)$ stating that $z$ is the pair $(x,y)$.

Stefan Mesken
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