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Every proper class is in bijection with the universe. Over class-theoretic Zermelo set theory, with class-parameter separation, this is equivalent to replacement plus global choice. For replacement, a proper functional image of a set would give a surjection from that set onto the universe, contradicted by diagonal separation. The proper class of ordinals supplies the global well-order. Conversely rank followed by uniformly chosen rank-segment well-orders is set-like, and every proper class in that order has order type equal to the entire ordinal class.

Interpret the assertion in the usual well-founded, class-theoretic version of Zermelo set theory, including foundation, class comprehension and separation with class parameters. Classes are not extra sets. The axiom of limitation of size, in the form used here, says that every proper class is in bijection with the universe $V$.

First derive replacement. Let a functional class relation map a set $a$ onto its image class $B$. If $B$ were proper, the limitation axiom would give a bijection $h:B\to V$. Its composite with the original map would be a surjection $g:a\to V$. Class-parameter separation formsSurjectivity gives $d\in a$ with $g(d)=D$, whence $d\in D$ iff $d\notin D$, a contradiction. Thus $B$ is a set, proving each instance of replacement. The argument also gives class replacement when the functional class is a parameter.

The class $\mathrm{On}$ of ordinals is proper: were it a set, its union and successor would be ordinals larger than every ordinal in it. Limitation gives a bijection $e:\mathrm{On}\to V$. Ordering $V$ by the ordinal indices of $e$ gives a global well-order; each nonempty set has a member with least index. Choosing that member defines global choice.

Conversely assume replacement and global choice. Ordinary choice well-orders each rank segment, and global choice selects such a well-order uniformly, since the collection of well-orders of each fixed rank segment is a nonempty set. Order all sets first by their rank of a set and then by the selected order within that rank. The resulting global order is set-like: the predecessors of any element are contained in a rank segment and form a set. Restrict it to a proper class $A$. By transfinite recursion, enumerate $A$ in this order. It cannot be exhausted at a set ordinal, since replacement would then make $A$ a set. Every member appears, because its predecessors form a set, whose well-order has an ordinal order type. Hence $A$ has order type $\mathrm{On}$. The same applies to $V$, and composing the two enumerations gives a bijection $A\cong V$. Thus limitation of size is equivalent to replacement plus global choice in this class-theoretic setting.