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Theorem peano3 7129
 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 5843 . 2 suc 𝐴 ≠ ∅
21a1i 11 1 (𝐴 ∈ ω → suc 𝐴 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2030   ≠ wne 2823  ∅c0 3948  suc csuc 5763  ωcom 7107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-suc 5767 This theorem is referenced by: (None)
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