elementary set theory - Proving $(Ale B)vee (Ble A)$ for sets $A$ and $B$

For any pair of sets $A$ and $B$, we can define $A\le B$ iff there exists injection $f\colon A\rightarrow B$. I am trying prove that $$(A\le B)\vee (B\le A).$$

I have tried assuming $\neg (A\le B)$, then proving $B\le A$ by constructing the required injection, but I haven't been able to make any progress. Any hints, etc. would be appreciated.

EDIT

Assuming $\neg (A\le B)$, can you prove there exists a surjection $f: A\rightarrow B$? Then it would be easy, by applying AC, to construct an injection $g: B\rightarrow A$