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 $0^{\sharp}$? 



Is the inclusion version of Kunen inconsistency theorem true?
$ \newcommand\ofnoteq{\subsetneq}$


It is a very nice question!


The answer is that there are numerous definable automorphisms of $\langle V,\ofnoteq\rangle$. To see this, let $f:V\to V$ be any permutation of the universe, and define the induced function $\pi:V\to V$ by $\pi(x)=f[x]$, the image of $x$ under $f$. For example, $\pi$ maps the singleton $\{ a\}$ to the singleton $\{f(a)\}$. Using the fact that $f$ is a permutation, it is not difficult to see that $x\ofnoteq y\iff \pi(x)\ofnoteq\pi(y)$, and furthermore that $\pi$ is a bijection, and hence it is an automorphism of the universe with respect to $\ofnoteq$, and consequently an elementary embedding, which is nontrivial precisely when $f$ is.


So there is no large cardinal strength here to be found, and the lesson appears to be that $\ofnoteq$ does not capture much of the intended set-theoretic structure. 


