Recall the definition of critical point for set theory:

A critical point of an elementary embedding of one transitive class into another transitive class is the smallest ordinal not mapped to itself. (This is from the Wikipedia article "Critical point (set theory)", which claims this definition is from Jech's Set Theory (2002 edition)).

What theorems about critical points of elementary embeddings can be proven in $ZF$ without recourse to the Axiom of Choice? To be specific, are these theorems enough to prove anything useful regarding critical points of nontrivial elementary embeddings $j$: $V$$\rightarrow$$V$?