Consider the following statement
$(C_2)$: For every set $X$ there exists a compact Hausdorff topology on $X$.
$C_2$ is a theorem of $ZFC$, because Choice gives us a bijection between any infinite set and the successor ordinal of the cardinality of the set, and any successor ordinal is compact and Hausdorff in its order topology.
Question: Is $C_2$ a theorem of $ZF$?