Theorem peano3 4632

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 4453	1 |- ( A e. om -> suc A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2167   =/= wne 2367   (/)c0 3450   suc csuc 4400   omcom 4626

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 3451  df-sn 3628  df-suc 4406

This theorem is referenced by:  nndceq0  4654  frecabcl  6457  frecsuclem  6464  nnsucsssuc  6550  php5  6919  findcard2  6950  findcard2s  6951  omp1eomlem  7160  ctmlemr  7174  nnsf  15649  peano4nninf  15650