Theorem peano3 4694

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 4515	1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2202   ≠ wne 2402  ∅c0 3494  suc csuc 4462  ωcom 4688

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-un 3204  df-nul 3495  df-sn 3675  df-suc 4468

This theorem is referenced by:  nndceq0  4716  frecabcl  6564  frecsuclem  6571  nnsucsssuc  6659  php5  7043  findcard2  7077  findcard2s  7078  omp1eomlem  7292  ctmlemr  7306  nnsf  16607  peano4nninf  16608