Theorem elni 5004

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 5000	. . 3 |- N. = (om \ {(/)})
2	1	eleq2i 1538	. 2 |- (A e. N. <-> A e. (om \ {(/)}))
3		eldifsn 2462	. 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 958   =/= wne 1585   \ cdif 2044  (/)c0 2280  {csn 2409  omcom 3131  N.cnpi 4972

This theorem is referenced by:  elni2 5005  0npi 5010  1pi 5011  addclpi 5020  mulclpi 5021  nlt1pi 5033  indpi 5034

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-sn 2412  df-pr 2413  df-ni 5000

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