The relations $\in$ and $\subsetneq$ seem so similar in some sense. For example they are equal on ordinal numbers. So there is a natural question about their possible similar behaviors on the constructible universe or proper class of all sets for example in the case of Kunen inconsistency theorem.

**Question (1):** Is there a non-trivial elementary embedding $j:\langle V,\subsetneq\rangle \longrightarrow \langle V,\subsetneq\rangle$?

**Question (2):** Is there a non-trivial elementary embedding $j:\langle L,\subsetneq\rangle \longrightarrow \langle L,\subsetneq\rangle$?

**Question (3):** What is the consistency strength of existence of such embeddings relative to large cardinal axioms? Particularly what is the position of existence of a non-trivial elementary embedding from $\langle L,\subsetneq\rangle$ to itself relative to existence of $0^{\sharp}$?

verydifferent. The former is well-founded and the latter is not. $\endgroup$ – Asaf Karagila Oct 7 '13 at 21:30