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Theorem peano3 4244
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 𝜔 → suc A ≠ ∅)

Proof of Theorem peano3
StepHypRef Expression
1 nsuceq0g 4102 1 (A 𝜔 → suc A ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1375  wne 2187  c0 3200  suc csuc 4049  𝜔com 4238
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 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  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 2896  df-un 2898  df-nul 3201  df-sn 3355  df-suc 4055
This theorem is referenced by:  nndceq0  4264  frecsuclem3  5901  nnsucsssuc  5981
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