A well-known result of Kunen says that there is no non-trivial elementary embedding $j: V \rightarrow V.$ There are several proofs of this theorem (see Kanamori, The higher infinite). I wonder to know if the following argument works:


Assume on the contrary that such an embedding exists. Let $\Phi(\alpha)$ be the following statement:


'' $\alpha=crit(j)$ for some non-trivial $j: V \rightarrow V.$''


Let $\kappa$ be the least such that $\Phi(\kappa)$ and let $j: V \rightarrow V$ witness this. Using $j$ we can conclude that $j(\kappa)$ is also the least such that $\Phi(j(\kappa))$ and this is impossible.



Kunen's inconsistency result
Your $\Phi(\kappa)$ uses an existential quantifier over *classes* (since $j$ is a class), whereas $j$ is only assumed to preserve *first order* formulae, so there is no reason to conclude that $j(\kappa)$ is also the least such that $\Phi(j(\kappa))$,


