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Theorem peano3 7898
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 6454 . 2 suc 𝐴 ≠ ∅
21a1i 11 1 (𝐴 ∈ ω → suc 𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wne 2929  c0 4322  suc csuc 6373  ωcom 7871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-v 3463  df-dif 3947  df-un 3949  df-nul 4323  df-sn 4631  df-suc 6377
This theorem is referenced by: (None)
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