Consider the three following large cardinal axioms:

- there exists a nontrivial elementary embedding $j:V\to V$.
- there exists a n.e.e. $j:V\to M$ such that $M^{j^\omega(crit(j))}\subseteq M$.
- there exists a n.e.e. $j:V_{\lambda+2}\to V_{\lambda+2}$ for some $\lambda$.

Kunen's inconsistency theorem states that, in ZFC (with replacement extended to formulas containing $j$), these three statements are false. However there is no such (known) result within ZF (with replacement sufficiently extended) alone.

Clearly (1) implies (2). I remember seeing somewhere that (2) implies (1), but I don't remember where, so I might be mistaken. Does anyone knows how these three statements relates to each other?

EDIT: this question already deals with the case (1)=>(3)