Metamath Proof Explorer

Definition df-suc

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. 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)

Ref Expression
Assertion df-suc ${⊢}\mathrm{suc}{A}={A}\cup \left\{{A}\right\}$

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ${class}{A}$
1 0 csuc ${class}\mathrm{suc}{A}$
2 0 csn ${class}\left\{{A}\right\}$
3 0 2 cun ${class}\left({A}\cup \left\{{A}\right\}\right)$
4 1 3 wceq ${wff}\mathrm{suc}{A}={A}\cup \left\{{A}\right\}$