MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elni Structured version   Visualization version   GIF version

Theorem elni 10916
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 10912 . . 3 N = (ω ∖ {∅})
21eleq2i 2833 . 2 (𝐴N𝐴 ∈ (ω ∖ {∅}))
3 eldifsn 4786 . 2 (𝐴 ∈ (ω ∖ {∅}) ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
42, 3bitri 275 1 (𝐴N ↔ (𝐴 ∈ ω ∧ 𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2108  wne 2940  cdif 3948  c0 4333  {csn 4626  ωcom 7887  Ncnpi 10884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-sn 4627  df-ni 10912
This theorem is referenced by:  elni2  10917  0npi  10922  1pi  10923  addclpi  10932  mulclpi  10933  nlt1pi  10946  indpi  10947
  Copyright terms: Public domain W3C validator