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Theorem peano3 6791
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 5554 . 2 suc 𝐴 ≠ ∅
21a1i 11 1 (𝐴 ∈ ω → suc 𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1937  wne 2675  c0 3757  suc csuc 5476  ωcom 6769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-nul 4567
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-v 3068  df-dif 3429  df-un 3431  df-nul 3758  df-sn 3996  df-suc 5480
This theorem is referenced by: (None)
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