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I believe that "The continuum is a limit cardinal" does the job. This can be expressed in a $\Pi^2_2$ way as "For every set of reals $X$, either there is a surjection $X\rightarrow\mathbb{R}$ or there is a set of reals $Y$ such that there is no surjection $X\rightarrow Y$ or $Y\rightarrow\mathbb{R}$," and I don't see how to get a $\Sigma^2_2$ equivalent even granting large cardinals. (In particular, note that "The continuum is greater than or equal to some uncountable limit cardinal" is easy to express in a $\Sigma^2_2$ way as "There exists an $\omega$-sequence of sets of reals of strictly increasing cardinality," but this doesn't seem to be useful here.)