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Theorem peano3 4580
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 4403 1  |-  ( A  e.  om  ->  suc  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2141    =/= wne 2340   (/)c0 3414   suc csuc 4350   omcom 4574
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-un 3125  df-nul 3415  df-sn 3589  df-suc 4356
This theorem is referenced by:  nndceq0  4602  frecabcl  6378  frecsuclem  6385  nnsucsssuc  6471  php5  6836  findcard2  6867  findcard2s  6868  omp1eomlem  7071  ctmlemr  7085  nnsf  14038  peano4nninf  14039
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