User:Dfeuer/Cartesian Product of Sets is Set

Theorem

Let $a$ and $b$ be sets.

Then the Cartesian product $a \times b$ is a set.

Proof

By User:Dfeuer/Cartesian Product is Subclass of Power Set of Power Set of Union:

$a \times b \subseteq \mathcal P(\mathcal P(a \cup b))$

By User:Dfeuer/Binary Union of Sets is Set, the union axiom, the power set axiom, and the subset axiom, $a \times b$ is a set.

$\blacksquare$