Theorem pinn 10769

Description: A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
pinn	⊢ (𝐴 ∈ N → 𝐴 ∈ ω)

Proof of Theorem pinn

Step	Hyp	Ref	Expression
1		df-ni 10763	. . 3 ⊢ N = (ω ∖ {∅})
2		difss 4086	. . 3 ⊢ (ω ∖ {∅}) ⊆ ω
3	1, 2	eqsstri 3981	. 2 ⊢ N ⊆ ω
4	3	sseli 3930	1 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)

Syntax hints:   → wi 4   ∈ wcel 2111   ∖ cdif 3899  ∅c0 4283  {csn 4576  ωcom 7796  Ncnpi 10735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3905  df-ss 3919  df-ni 10763

This theorem is referenced by:  pion  10770  piord  10771  mulidpi  10777  addclpi  10783  mulclpi  10784  addcompi  10785  addasspi  10786  mulcompi  10787  mulasspi  10788  distrpi  10789  addcanpi  10790  mulcanpi  10791  addnidpi  10792  ltexpi  10793  ltapi  10794  ltmpi  10795  indpi  10798