MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  peano3 Unicode version

Theorem peano3 6630
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

Proof of Theorem peano3
StepHypRef Expression
1 nsuceq0 4916 . 2
21a1i 11 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  e.wcel 1758  =/=wne 2648   c0 3751  succsuc 4838   com 6609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4538
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3083  df-dif 3445  df-un 3447  df-nul 3752  df-sn 3994  df-suc 4842
  Copyright terms: Public domain W3C validator