Theorem peano3

Description: The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of TakeutiZaring p. 42. (Contributed by NM, 3-Sep-2003) Avoid ax-nul . (Revised by Umit Teoman Dogan, 10-Jun-2026)

	Ref	Expression
Assertion	peano3	$$⊢A\in \mathrm{\omega }\to \mathrm{suc}A\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		sucidg	$$⊢A\in \mathrm{\omega }\to A\in \mathrm{suc}A$$
2	1	ne0d	$$⊢A\in \mathrm{\omega }\to \mathrm{suc}A\ne \varnothing$$