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Theorem peano3 4557
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
StepHypRef Expression
1 nsuceq0g 4380 1  |-  ( A  e.  om  ->  suc  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2128    =/= wne 2327   (/)c0 3395   suc csuc 4327   omcom 4551
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-dif 3104  df-un 3106  df-nul 3396  df-sn 3567  df-suc 4333
This theorem is referenced by:  nndceq0  4579  frecabcl  6348  frecsuclem  6355  nnsucsssuc  6441  php5  6805  findcard2  6836  findcard2s  6837  omp1eomlem  7040  ctmlemr  7054  nnsf  13648  peano4nninf  13649
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