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Theorem pinn 9987
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 9981 . . 3 N = (ω ∖ {∅})
2 difss 3934 . . 3 (ω ∖ {∅}) ⊆ ω
31, 2eqsstri 3830 . 2 N ⊆ ω
43sseli 3793 1 (𝐴N𝐴 ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  cdif 3765  c0 4114  {csn 4367  ωcom 7298  Ncnpi 9953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-v 3386  df-dif 3771  df-in 3775  df-ss 3782  df-ni 9981
This theorem is referenced by:  pion  9988  piord  9989  mulidpi  9995  addclpi  10001  mulclpi  10002  addcompi  10003  addasspi  10004  mulcompi  10005  mulasspi  10006  distrpi  10007  addcanpi  10008  mulcanpi  10009  addnidpi  10010  ltexpi  10011  ltapi  10012  ltmpi  10013  indpi  10016
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