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

Theorem peano3 6979
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 nsuceq0 5722 . 2 suc 𝐴 ≠ ∅
21a1i 11 1 (𝐴 ∈ ω → suc 𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  wne 2780  c0 3874  suc csuc 5642  ωcom 6957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-suc 5646
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator