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Theorem peano3 4478
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 4308 1  |-  ( A  e.  om  ->  suc  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1463    =/= wne 2283   (/)c0 3331   suc csuc 4255   omcom 4472
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-v 2660  df-dif 3041  df-un 3043  df-nul 3332  df-sn 3501  df-suc 4261
This theorem is referenced by:  nndceq0  4499  frecabcl  6262  frecsuclem  6269  nnsucsssuc  6354  php5  6718  findcard2  6749  findcard2s  6750  omp1eomlem  6945  ctmlemr  6959  nnsf  13033  peano4nninf  13034
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