Theorem elni 5069

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 5065	. . 3 |- N. = (om \ {(/)})
2	1	eleq2i 1585	. 2 |- (A e. N. <-> A e. (om \ {(/)}))
3		eldifsn 2516	. 2 |- (A e. (om \ {(/)}) <-> (A e. om /\ A =/= (/)))
4	2, 3	bitri 180	1 |- (A e. N. <-> (A e. om /\ A =/= (/)))

Syntax hints:   <-> wb 153   /\ wa 230   e. wcel 999   =/= wne 1632   \ cdif 2095  (/)c0 2331  {csn 2461  omcom 3188  N.cnpi 5037

This theorem is referenced by:  elni2 5070  0npi 5075  1pi 5076  addclpi 5085  mulclpi 5086  nlt1pi 5098  indpi 5099

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-v 1859  df-dif 2100  df-un 2101  df-sn 2464  df-pr 2465  df-ni 5065

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