Theorem pinn 10778

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 10772	. . 3 ⊢ N = (ω ∖ {∅})
2		difss 4085	. . 3 ⊢ (ω ∖ {∅}) ⊆ ω
3	1, 2	eqsstri 3977	. 2 ⊢ N ⊆ ω
4	3	sseli 3926	1 ⊢ (𝐴 ∈ N → 𝐴 ∈ ω)

Syntax hints:   → wi 4   ∈ wcel 2113   ∖ cdif 3895  ∅c0 4282  {csn 4577  ωcom 7804  Ncnpi 10744

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 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-dif 3901  df-ss 3915  df-ni 10772

This theorem is referenced by:  pion  10779  piord  10780  mulidpi  10786  addclpi  10792  mulclpi  10793  addcompi  10794  addasspi  10795  mulcompi  10796  mulasspi  10797  distrpi  10798  addcanpi  10799  mulcanpi  10800  addnidpi  10801  ltexpi  10802  ltapi  10803  ltmpi  10804  indpi  10807