= Axiom of limitation of size
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= Von Neumann axiom of limitation of size
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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.
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