For Kunen set theory i.10.19(4)
$\alpha\neq 0$, $\alpha$ is a successor ordinal if and only if $\omega_\alpha$ is a successor cardinal
The forward direction seems more or less straight forward since if $\alpha$ is a successor ordinal, we have some $\beta$ such that $\alpha = \beta + 1$. Then, we have $\omega_\beta^+ = \omega_{\beta + 1} = \omega_\alpha$ by definition and $\omega_\alpha$ is a successor cardinal.
I'm having trouble with the other direction. My thought is to use the contrapositive. Say we have $\alpha$ to be a limit ordinal ($\alpha\neq 0$), then $\omega_\alpha\neq\omega$. Assume $\omega_\alpha$ is a successor cardinal, then we can find $\beta$ such that $\omega_\beta^+ \omega_\alpha$. Then, we should have $\beta + 1 = \alpha$ so $\alpha$ is a successor ordinal. This gives a contradiction.
Will the proof for converse direction work? I'm not too comfortable picking $\beta$ since there isn't exactly ensures the existance of $\beta$.
Sometimes it's good to work in the positive and just chase down the definitions.
If $\omega_\alpha$ is a successor cardinal, then it is the successor of some cardinal. If you've shown that every infinite cardinal is of the form $\omega_\beta$, then that predecessor cardinal is such $\omega_\beta$.