= Solution
Apply the stated <cardinal comparability principle> to $x$ and its Hartogs ordinal $h(x)$. The <Hartogs theorem> rules out an injection $h(x)\to x$, so comparability supplies an injection
$$
j:x\longrightarrow h(x).
$$
Order $x$ by $u\prec v$ exactly when $j(u)\in j(v)$. This is the restriction of an ordinal well-order and therefore well-orders $x$. Hence every set can be well-ordered. The <well-ordering theorem> is equivalent to the <axiom of choice>: for a family of nonempty sets, well-order its union and choose the least member of each set. The comparison statement therefore implies choice.
Back to article page
