= Diagonal argument for ultrapower cardinality
{title2=$|M^\kappa/U|>\kappa$}
If $U$ is a <kappa-complete> <nonprincipal ultrafilter> on $\kappa$ and $|M|\geq\kappa$, then $|M^\kappa/U|>\kappa$. Otherwise enumerate representatives $f_\alpha$ for its classes and choose $g(i)\notin\{f_\alpha(i):\alpha<i\}$. Every tail of $\kappa$ belongs to $U$, so $g$ differs modulo $U$ from every representative. The <small set is absent from a complete nonprincipal ultrafilter> lemma supplies the tail property.
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