We have the following results
Let $A$ and $B$ be infinite sets s.t. $|A|=|B|$, then $|A\cup B|=|A|$.
I was wondering if we can prove that without the Axiom of Choice or without using cardinal numbers, it's showing that exists the bijection. I'm trying the second part using Zorn's lemma but I've failed.
In the other hand, we have
Let $A$ be an infinite set, then $|A\times A|=|A|$.
I know this is equivalent to Axiom of Choice, but I'm trying a proof without using cardinal numbers.
Please, could you help me whit ideas for that?