Theorem peano3 4743

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 4563	1 |- ( A e. om -> suc A =/= (/) )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   =/= wne 2420   (/)c0 3520   suc csuc 4510   omcom 4737

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-sn 3715  df-suc 4516

This theorem is used by:  nndceq0  4765  frecabcl  6670  frecsuclem  6677  nnsucsssuc  6765  php5  7159  findcard2  7193  findcard2s  7194  omp1eomlem  7434  ctmlemr  7448  nnsf  17048  peano4nninf  17049