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.