Show that the well-ordering theorem is equivalent to:
For any two sets $X$ and $Y$ there is an injection of $X$ to $Y$ or $Y$ to $X$.
I have only some ideas for the "$\Leftarrow$" direction:
If we have $f:X\to Y$ is injective, then follows ((5.2) in Jech) that $|X|\geq |Y|$.
(By choosing $f^{-1}(\{y\})$ for each $y\in Y$.)
Since we can choose $X$ and $Y$ arbitrarily, I thought we can then compare all elements in $X$ and also find a least element. But I don't really know how to get there formally.
For the other direction I have no idea yet.
I'd be very happy for any hint how to approach this proof!
Hint: For a set $X$ show that there exists a least ordinal $\kappa$ such that $\kappa\nleq|X|$. In order to show its existence you will need to show it is the range of a function whose domain is $\mathcal{P(P(}X))$ (and use replacement).
Now we have that $\kappa\nleq|X|$ so $|X|<\kappa$. Conclude Zermelo's theorem.