Theorem pinn 10634

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 10628	. . 3 ⊢ N = (ω ∖ {∅})
2		difss 4066	. . 3 ⊢ (ω ∖ {∅}) ⊆ ω
3	1, 2	eqsstri 3955	. 2 ⊢ N ⊆ ω
4	3	sseli 3917	1 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)

Syntax hints:   → wi 4   ∈ wcel 2106   ∖ cdif 3884  ∅c0 4256  {csn 4561  ωcom 7712  Ncnpi 10600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-ni 10628

This theorem is referenced by:  pion  10635  piord  10636  mulidpi  10642  addclpi  10648  mulclpi  10649  addcompi  10650  addasspi  10651  mulcompi  10652  mulasspi  10653  distrpi  10654  addcanpi  10655  mulcanpi  10656  addnidpi  10657  ltexpi  10658  ltapi  10659  ltmpi  10660  indpi  10663