Quote from Wikipedia, infinite set:
The Cartesian product of an infinite number of sets each containing at least two elements is either empty or infinite.
I know that this is the regime where we need AC, but before that, ...
How can I prove the quoted statement above?
You don't need the axiom of choice to prove this statement. You need the axiom of choice to prove that the product is non-empty, but here you allow for that possibility.
Let $\{P_i\mid i\in I\}$ be an infinite family of sets with at least two elements each. If $\prod P_i$ is empty we're done, if not there is at least one $F$ in the product. Define a function from the product onto $I$ as follows:
Let $f$ be in the product, if $f$ and $F$ differ on exactly one coordinate, $i$, then map $f$ to $i$. Otherwise map $f$ to some fixed $i_0\in I$.
You can show now that this function is surjective. And since $I$ is infinite, the product must be infinite too (since the image of a finite set is finite).