Theorem elni 6156

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 6152	. . 3 |- N. = (om \ {(/)})
2	1	eleq2i 1961	. 2 |- (A e. N. <-> A e. (om \ {(/)}))
3		eldifsn 3123	. 2 |- (A e. (om \ {(/)}) <-> (A e. om /\ A =/= (/)))
4	2, 3	bitri 190	1 |- (A e. N. <-> (A e. om /\ A =/= (/)))

Syntax hints:   <-> wb 163   /\ wa 240   e. wcel 1300   =/= wne 2017   \ cdif 2590  (/)c0 2875  {csn 3044  omcom 3949  N.cnpi 6124

This theorem is referenced by:  elni2 6157  0npi 6162  1pi 6163  addclpi 6172  mulclpi 6173  nlt1pi 6185  indpi 6186

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-sn 3049  df-ni 6152

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