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Theorem pinn 10916
Description: A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
pinn (𝐴N𝐴 ∈ ω)

Proof of Theorem pinn
StepHypRef Expression
1 df-ni 10910 . . 3 N = (ω ∖ {∅})
2 difss 4146 . . 3 (ω ∖ {∅}) ⊆ ω
31, 2eqsstri 4030 . 2 N ⊆ ω
43sseli 3991 1 (𝐴N𝐴 ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cdif 3960  c0 4339  {csn 4631  ωcom 7887  Ncnpi 10882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-ss 3980  df-ni 10910
This theorem is referenced by:  pion  10917  piord  10918  mulidpi  10924  addclpi  10930  mulclpi  10931  addcompi  10932  addasspi  10933  mulcompi  10934  mulasspi  10935  distrpi  10936  addcanpi  10937  mulcanpi  10938  addnidpi  10939  ltexpi  10940  ltapi  10941  ltmpi  10942  indpi  10945
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