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

Theorem peano3 4825
Description: The successor of any natural number is not zero. One of Peano's 5 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 nsuceq0 4621 . 2  |-  suc  A  =/=  (/)
21a1i 11 1  |-  ( A  e.  om  ->  suc  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721    =/= wne 2567   (/)c0 3588   suc csuc 4543   omcom 4804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-un 3285  df-nul 3589  df-sn 3780  df-suc 4547
  Copyright terms: Public domain W3C validator