Doubts about cartesian product existence.

Question

Why I can't to show that the cartesian product between two sets exists without replacement or power set axioms?

Answer 1

There is a unique empty set (Existence, Extensionality, Comprehension).

For any $x,y$ there is a unique $z=\{\{x\},\{x,y\}\}=(x,y)$ (Pairing, Extensionality).

For any $X,Y$ there exists $X\cup Y=\bigcup \{X,Y\}$ (Pairing, Union).

Let $g(x,y)$ be $(x\in X\land y\in Y).$

Let $f(x,y,z)$ be $(\;(g(x,y)\land z=(x,y))\lor ((\neg g(x,y)\land z=\emptyset)\;).$

Apply Replacement & Comprehension to $\forall x\in X\cup Y\;\forall y\in X\cup Y\;\exists! z\;(f(x,y,z)$ to obtain, for any $X,Y$ that $$\exists Z\;\forall z\;(z\in Z\iff \exists x\in X\cup Y\;\exists y\in X\cup Y\;(f(x,y,z)).$$ For such $Z$ we have, by Comprehension, the existence of $\{z\in Z: z\ne \emptyset\},$ which is $X\times Y.$