Theorem peano3 4692

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 4513	1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2200   ≠ wne 2400  ∅c0 3492  suc csuc 4460  ωcom 4686

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2802  df-dif 3200  df-un 3202  df-nul 3493  df-sn 3673  df-suc 4466

This theorem is referenced by:  nndceq0  4714  frecabcl  6560  frecsuclem  6567  nnsucsssuc  6655  php5  7039  findcard2  7071  findcard2s  7072  omp1eomlem  7284  ctmlemr  7298  nnsf  16543  peano4nninf  16544