Theorem pinn 10801

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 10795	. . 3 ⊢ N = (ω ∖ {∅})
2		difss 4076	. . 3 ⊢ (ω ∖ {∅}) ⊆ ω
3	1, 2	eqsstri 3968	. 2 ⊢ N ⊆ ω
4	3	sseli 3917	1 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)

Syntax hints:   → wi 4   ∈ wcel 2114   ∖ cdif 3886  ∅c0 4273  {csn 4567  ωcom 7817  Ncnpi 10767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-dif 3892  df-ss 3906  df-ni 10795

This theorem is referenced by:  pion  10802  piord  10803  mulidpi  10809  addclpi  10815  mulclpi  10816  addcompi  10817  addasspi  10818  mulcompi  10819  mulasspi  10820  distrpi  10821  addcanpi  10822  mulcanpi  10823  addnidpi  10824  ltexpi  10825  ltapi  10826  ltmpi  10827  indpi  10830