$(X,A)$, $(Y,B)$, $(X \times Y, A \otimes B)$ measurable spaces. How to show $A=\{\emptyset, X\}$ or $B=\{\emptyset, Y\}$?

Question

like the title said, I want to show that $A=\{\emptyset, X\}$ or $B=\{\emptyset, Y\}$ if $(X,A)$, $(Y,B)$, $(X \times Y, A \otimes B)$ are measurable spaces.

I tried to form a contradiction by assuming there exists an element neither $\emptyset$ nor X in A but it didn't work or I just didn't understand it.

If you could help me I would appreciate it.