Theorem peano3 7902

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 5259. (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 6446	. 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴)
2	1	ne0d 4287	1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ≠ wne 2955  ∅c0 4278  suc csuc 6364  ωcom 7877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4584  df-suc 6368

This theorem is used by: (None)