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Theorem peano3 4222
Description: The successor of any natural number is not zero. One of Peano's 5 postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
Assertion
Ref Expression
peano3  om  suc  =/=  (/)

Proof of Theorem peano3
StepHypRef Expression
1 nsuceq0g 4080 1  om  suc  =/=  (/)
Colors of variables: wff set class
Syntax hints:   wi 4   wcel 1376    =/= wne 2187   (/)c0 3203   suc csuc 4028   omcom 4216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1232  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ne 2189  df-v 2536  df-dif 2899  df-un 2901  df-nul 3204  df-sn 3334  df-suc 4033
This theorem is referenced by:  nndceq0  4242  frecsuclem3  5881  nnsucsssuc  5960
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