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Theorem peano3 6854
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 5610 . 2 suc 𝐴 ≠ ∅
21a1i 11 1 (𝐴 ∈ ω → suc 𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1938  wne 2684  c0 3777  suc csuc 5532  ωcom 6832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-nul 4616
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-v 3079  df-dif 3447  df-un 3449  df-nul 3778  df-sn 4029  df-suc 5536
This theorem is referenced by: (None)
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