Kunen inconsistency theorem

In ZFC there is no nonidentity elementary embedding $j:V\to V$, understood as an ordinary class embedding whose set restrictions and critical sequence are available. Let $\lambda$ be the supremum of the Kunen critical sequence, so $j(\lambda )=\lambda$ and $j({\lambda }^{+})={\lambda }^{+}$. Partition the ordinals of cofinality $\omega$ below ${\lambda }^{+}$ into stationary sets indexed by the critical point. After applying $j$, the extra stationary cell at that critical point meets the club of ordinals closed under $j$. Such an ordinal of countable cofinality is fixed by $j$ and hence belongs both to that extra cell and an old-index cell, a contradiction.