Hartogs' Theorem states: If $X$ is a set, then there is an ordinal $\alpha$ such that there is no injection from $\alpha$ to $X$. The proof I am familiar with uses unique representation of wellorders by ordinals, which in turn requires the Axiom Scheme of Replacement. How do you prove the following without using Replacement or the Axiom of Choice?
For every set there is a wellorder such that there is no injection from the wellorder to the set.
I had the idea to separate from $\{X\}\times \mathcal P(X \times X)$ the set of all wellorders on $X$ and then to extend this wellorder, but as far as I understand, without the Axiom of Choice we do not know whether there even exists a wellorder on $X$.
