Theorem peano3 4573

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 e. om -> suc A =/= (/) )

Proof of Theorem peano3

Step	Hyp	Ref	Expression
1		nsuceq0g 4396	1 |- ( A e. om -> suc A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2136   =/= wne 2336   (/)c0 3409   suc csuc 4343   omcom 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-dif 3118  df-un 3120  df-nul 3410  df-sn 3582  df-suc 4349

This theorem is referenced by:  nndceq0  4595  frecabcl  6367  frecsuclem  6374  nnsucsssuc  6460  php5  6824  findcard2  6855  findcard2s  6856  omp1eomlem  7059  ctmlemr  7073  nnsf  13895  peano4nninf  13896