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Rank-into-rank

In set theory, the following four axioms, commonly known as rank-into-rank embeddings, are among the most powerful large cardinal axioms known. Stated in the order of increasing strength, they are as follows:

I3: There is a nontrivial elementary embedding of Vλ into itself.

I2: There is a nontrivial elementary embedding of V into a transitive class M that includes Vλ where λ is the first fixed point above the critical point.

I1: There is a nontrivial elementary embedding of Vλ+1 into itself.

I0: There is a nontrivial elementary embedding of L(Vλ+1 ) into itself with the critical point below λ.

Assuming the axiom of choice, it is provable that if there is a nontrivial elementary embedding of Vλ into itself then λ is a limit ordinal of cofinality ω or the successor of such an ordinal.