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Theorem elni 7309
Description: Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.)
Assertion
Ref Expression
elni (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))

Proof of Theorem elni
StepHypRef Expression
1 df-ni 7305 . . 3 N = (ω ∖ {∅})
21eleq2i 2244 . 2 (𝐴N𝐴 ∈ (ω ∖ {∅}))
3 eldifsn 3721 . 2 (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
42, 3bitri 184 1 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wcel 2148  wne 2347  cdif 3128  c0 3424  {csn 3594  ωcom 4591  Ncnpi 7273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2741  df-dif 3133  df-sn 3600  df-ni 7305
This theorem is referenced by:  0npi  7314  elni2  7315  1pi  7316  addclpi  7328  mulclpi  7329  nlt1pig  7342  indpi  7343  nqnq0pi  7439  prarloclemcalc  7503
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