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Theorem elni 7080
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 7076 . . 3 N = (ω ∖ {∅})
21eleq2i 2182 . 2 (𝐴N𝐴 ∈ (ω ∖ {∅}))
3 eldifsn 3618 . 2 (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
42, 3bitri 183 1 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 1463  wne 2283  cdif 3036  c0 3331  {csn 3495  ωcom 4472  Ncnpi 7044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-v 2660  df-dif 3041  df-sn 3501  df-ni 7076
This theorem is referenced by:  0npi  7085  elni2  7086  1pi  7087  addclpi  7099  mulclpi  7100  nlt1pig  7113  indpi  7114  nqnq0pi  7210  prarloclemcalc  7274
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