Theorem pinn 9700

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

Assertion

Ref	Expression
pinn	|- ( A e. N. -> A e. om )

Proof of Theorem pinn

Step	Hyp	Ref	Expression
1		df-ni 9694	. . 3 |- N. = ( om \ { (/) } )
2		difss 3737	. . 3 |- ( om \ { (/) } ) C_ om
3	1, 2	eqsstri 3635	. 2 |- N. C_ om
4	3	sseli 3599	1 |- ( A e. N. -> A e. om )

Syntax hints:   -> wi 4   e. wcel 1990   \ cdif 3571   (/)c0 3915   {csn 4177   omcom 7065   N.cnpi 9666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-ni 9694

This theorem is referenced by:  pion  9701  piord  9702  mulidpi  9708  addclpi  9714  mulclpi  9715  addcompi  9716  addasspi  9717  mulcompi  9718  mulasspi  9719  distrpi  9720  addcanpi  9721  mulcanpi  9722  addnidpi  9723  ltexpi  9724  ltapi  9725  ltmpi  9726  indpi  9729