# Proof of cartesian product being a set

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: $$(a,b) = \{\{a\},\{a,b\}\}$$

• This literally requires only the effort of writing down the definition of subset, power set, and union. – Asaf Karagila Apr 13 '18 at 11:05

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