I've seen an example of an elementary embedding such that $\omega_1$ is the critical point.

I was wondering what's wrong with the following proof that this cannot be:

Let $\phi(x_1,x_2)$ be the first-order formula: $\phi(x_1,x_2)=x_1\ is\ cardinal\wedge x_2\ is\ cardinal\wedge\nexists\kappa\left(\kappa\ is\ cardinal\wedge\ x_1<\kappa<x_2\right)$.

Let $j:M\to N$ be an elementary embedding with $crit(j)=\omega_1$, $j(\omega_1)=\kappa>\omega_1$

Then $M\models\phi(\omega_0,\omega_1)$, but $N\nvDash\phi(j(\omega_0),j(\omega_1))=\phi(\omega_0,\kappa)$

Contradiction to elementarity.

(By this [false] proof, $crit(j)$ cannot be a successive cardinal).

Thanks.