Description: Define the successor of a class. When applied to an ordinal number, the
successor means the same thing as "plus 1" (see oa1suc ). Definition
7.22 of TakeutiZaring p. 41, who use "+ 1" to denote this function.
Definition 1.4 of Schloeder p. 1, similarly. Ordinal natural numbers
defined using this successor function and 0 as the empty set are also
called von Neumann ordinals; 0 is the empty set {}, 1 is {0, {0}}, 2 is
{1, {1}}, and so on. Our definition is a generalization to classes.
Although it is not conventional to use it with proper classes, it has no
effect on a proper class ( sucprc ), so that the successor of any
ordinal class is still an ordinal class ( ordsuc ), simplifying certain
proofs. Some authors denote the successor operation with a prime
(apostrophe-like) symbol, such as Definition 6 of Suppes p. 134 and the
definition of successor in Mendelson p. 246 (who uses the symbol "Suc"
as a predicate to mean "is a successor ordinal"). The definition of
successor of Enderton p. 68 denotes the operation with a plus-sign
superscript. (Contributed by NM, 30-Aug-1993)