Theorem peano3 7890

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 5267. (Revised by Umit Teoman Dogan, 10-Jun-2026.)

Assertion

Ref	Expression
peano3	⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)

Proof of Theorem peano3

Step	Hyp	Ref	Expression
1		sucidg 6445	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴)
2	1	ne0d 4291	1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ≠ wne 2957  ∅c0 4282  suc csuc 6363  ωcom 7865

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3455  df-dif 3905  df-un 3907  df-nul 4283  df-sn 4588  df-suc 6367

This theorem is used by: (None)