Theorem peano2

Description: The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of TakeutiZaring p. 42. (Contributed by NM, 3-Sep-2003)

	Ref	Expression
Assertion	peano2	$$⊢A\in \mathrm{\omega }\to \mathrm{suc}A\in \mathrm{\omega }$$

Proof

Step	Hyp	Ref	Expression
1		peano2b	$$⊢A\in \mathrm{\omega }\leftrightarrow \mathrm{suc}A\in \mathrm{\omega }$$
2	1	biimpi	$$⊢A\in \mathrm{\omega }\to \mathrm{suc}A\in \mathrm{\omega }$$