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 ∈ ω → suc 𝐴 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		sucidg	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ suc 𝐴 )
2	1	ne0d	⊢ ( 𝐴 ∈ ω → suc 𝐴 ≠ ∅ )