Prove that $A \times B = B \times A $ if and only if $A = B \lor A = \emptyset \lor B = \emptyset$

Question

Prove that $A \times B = B \times A $ if and only if $A = B \lor A = \emptyset \lor B = \emptyset $
So, I need to prove that $$A\times B = B \times A \iff A = B \lor A = \emptyset \lor B = \emptyset$$ It is easy to show the right-to-left implication. The problem begins when I want to show the left-to-right one. I tried using the axiom of extentionality, but that did not work. What kind of trick should I use to prove that the relation from left to right holds as well?

Answer 1

Suppose $A \neq B \wedge A \neq \emptyset \wedge B \neq \emptyset$

Then $(\exists x \in A\wedge x \notin B) \vee (\exists x \in B\wedge x \notin A)$

We suppose WLoG $(\exists a \in A\wedge a \notin B)$

We also have $\exists b \in B$

This implies $$(a,b)\in A\times B$$ But $$(a,b)\notin B\times A$$ Because $$a\notin B$$

Therefore $A\times B\neq B\times A $