Theorem peano3 4633

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.)

Assertion

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

Proof of Theorem peano3

Step	Hyp	Ref	Expression
1		nsuceq0g 4454	1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2167   ≠ wne 2367  ∅c0 3451  suc csuc 4401  ωcom 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-un 3161  df-nul 3452  df-sn 3629  df-suc 4407

This theorem is referenced by:  nndceq0  4655  frecabcl  6466  frecsuclem  6473  nnsucsssuc  6559  php5  6928  findcard2  6959  findcard2s  6960  omp1eomlem  7169  ctmlemr  7183  nnsf  15736  peano4nninf  15737