Proof of cartesian product being a set

Question

How can I prove that this relation is correct (to prove of cartesian product being a set?).

$$ X \times Y \subset PP(X\cup Y) $$ I know it is based on the Kuratowsi's definition of an ordered pair, which should be: $$ (a,b) = \{\{a\},\{a,b\}\} $$

Answer 1

$$X \times Y = \{ (x,y): x\in X \text { & } y\in Y\}$$

Thus each element of $X\times Y $ is an ordered pair$$ (x,y) = \{\{x\},\{x,y\}\} $$

Note that $$\{\{x\},\{x,y\}\} \subseteq P(X\cup Y)$$

Thus $$\{\{x\},\{x,y\}\} \in PP(X\cup Y)$$

Which means $$X \times Y \subset PP(X\cup Y)$$