Prove that for any two sets $|X|$ and $|Y|$, either $|X|\leq|Y|$ or $|Y|\leq|X|$.

Question

Prove that for any two sets $|X|$ and $|Y|$, either $|X|\leq|Y|$ or $|Y|\leq|X|$.

I know that there is a proof using Zorn's Lemma but I can't figure out how to do it.

Answer 1

Let $C$ be the class of ordered pairs $(S,\phi)$ such that $S\subset X$ and $\phi:S\to Y$ is injective. Define a partial order on $C$ by saying $(S',\phi')\le(S,\phi)$ if $S'\subset S$ and $\phi|_{S'}=\phi'$.

Zorn's lemma shows that $C$ has a maximal element $(S,\phi)$. Now show that the maximality implies that either $S=X$ or $\phi(X)=Y$.