Theorem elni 5016

Description: Membership in the class of positive integers.

Assertion

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

Proof of Theorem elni

Step	Hyp	Ref	Expression
1		df-ni 5012	. . 3 |- N. = (om \ {(/)})
2	1	eleq2i 1541	. 2 |- (A e. N. <-> A e. (om \ {(/)}))
3		eldifsn 2466	. 2 |- (A e. (om \ {(/)}) <-> (A e. om /\ A =/= (/)))
4	2, 3	bitr 173	1 |- (A e. N. <-> (A e. om /\ A =/= (/)))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 960   =/= wne 1588   \ cdif 2047  (/)c0 2283  {csn 2413  omcom 3137  N.cnpi 4984

This theorem is referenced by:  elni2 5017  0npi 5022  1pi 5023  addclpi 5032  mulclpi 5033  nlt1pi 5045  indpi 5046

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-sn 2416  df-pr 2417  df-ni 5012

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