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- To prove the direction, let be a class and be a one-to-one correspondence from to Since maps onto the axiom of limitation of size implies that is a proper class.
To prove the direction, let be a proper class. We will define well-ordered classes and and construct order isomorphisms between and Then the order isomorphism from to is a one-to-one correspondence between and
It was proved above that the axiom of limitation of size implies that there is a function that maps onto Also, was defined as a subclass of that is a one-to-one correspondence between and It defines a well-ordering on if Therefore, is an order isomorphism from to
If is well-ordered class, its proper initial segments are the classes where Now has the property that all of its proper initial segments are sets. Since this property holds for The order isomorphism implies that this property holds for Since this property holds for
To obtain an order isomorphism from to the following theorem is used: If is a proper class and the proper initial segments of are sets, then there is an order isomorphism from to Since and satisfy the theorem's hypothesis, there are order isomorphisms and Therefore, the order isomorphism is a one-to-one correspondence between and (en)
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