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Theorem pinn 9902
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 9896 . . 3 N = (ω ∖ {∅})
2 difss 3888 . . 3 (ω ∖ {∅}) ⊆ ω
31, 2eqsstri 3784 . 2 N ⊆ ω
43sseli 3748 1 (𝐴N𝐴 ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  cdif 3720  c0 4063  {csn 4316  ωcom 7212  Ncnpi 9868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-ni 9896
This theorem is referenced by:  pion  9903  piord  9904  mulidpi  9910  addclpi  9916  mulclpi  9917  addcompi  9918  addasspi  9919  mulcompi  9920  mulasspi  9921  distrpi  9922  addcanpi  9923  mulcanpi  9924  addnidpi  9925  ltexpi  9926  ltapi  9927  ltmpi  9928  indpi  9931
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