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Theorem elni 10276
 Description: Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
elni (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Proof of Theorem elni
StepHypRef Expression
1 df-ni 10272 . . 3 N = (ω ∖ {∅})
21eleq2i 2902 . 2 (𝐴N𝐴 ∈ (ω ∖ {∅}))
3 eldifsn 4695 . 2 (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
42, 3bitri 277 1 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   ∈ wcel 2114   ≠ wne 3006   ∖ cdif 3910  ∅c0 4269  {csn 4543  ωcom 7558  Ncnpi 10244 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-ne 3007  df-v 3475  df-dif 3916  df-sn 4544  df-ni 10272 This theorem is referenced by:  elni2  10277  0npi  10282  1pi  10283  addclpi  10292  mulclpi  10293  nlt1pi  10306  indpi  10307
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