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Theorem peano3 4374
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 4209 1 (𝐴 ∈ ω → suc 𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  wne 2249  c0 3269  suc csuc 4156  ωcom 4368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-v 2614  df-dif 2986  df-un 2988  df-nul 3270  df-sn 3428  df-suc 4162
This theorem is referenced by:  nndceq0  4394  frecabcl  6096  frecsuclem  6103  nnsucsssuc  6185  php5  6504  findcard2  6535  findcard2s  6536
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