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Theorem peano3 4246
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  om  suc  =/=  (/)

Proof of Theorem peano3
StepHypRef Expression
1 nsuceq0g 4104 1  om  suc  =/=  (/)
Colors of variables: wff set class
Syntax hints:   wi 4   wcel 1374    =/= wne 2186   (/)c0 3201   suc csuc 4051   omcom 4240
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 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-suc 4057
This theorem is referenced by:  nndceq0  4266  frecsuclem3  5906  nnsucsssuc  5986
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