Theorem elni 9698

Description: Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
elni	|- ( A e. N. <-> ( A e. om /\ A =/= (/) ) )

Proof of Theorem elni

Step	Hyp	Ref	Expression
1		df-ni 9694	. . 3 |- N. = ( om \ { (/) } )
2	1	eleq2i 2693	. 2 |- ( A e. N. <-> A e. ( om \ { (/) } ) )
3		eldifsn 4317	. 2 |- ( A e. ( om \ { (/) } ) <-> ( A e. om /\ A =/= (/) ) )
4	2, 3	bitri 264	1 |- ( A e. N. <-> ( A e. om /\ A =/= (/) ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   \ 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-ne 2795  df-v 3202  df-dif 3577  df-sn 4178  df-ni 9694

This theorem is referenced by:  elni2  9699  0npi  9704  1pi  9705  addclpi  9714  mulclpi  9715  nlt1pi  9728  indpi  9729