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Theorem peano3 4635
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 4430 . 2  |-  suc  A  =/=  (/)
21a1i 12 1  |-  ( A  e.  om  ->  suc  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    e. wcel 1621    =/= wne 2419   (/)c0 3416   suc csuc 4352   omcom 4614
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-nul 4109
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-v 2759  df-dif 3116  df-un 3118  df-nul 3417  df-sn 3606  df-suc 4356
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