ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elni GIF version

Theorem elni 7306
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 7302 . . 3 N = (ω ∖ {∅})
21eleq2i 2244 . 2 (𝐴N𝐴 ∈ (ω ∖ {∅}))
3 eldifsn 3719 . 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 3126  c0 3422  {csn 3592  ωcom 4589  Ncnpi 7270
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 2739  df-dif 3131  df-sn 3598  df-ni 7302
This theorem is referenced by:  0npi  7311  elni2  7312  1pi  7313  addclpi  7325  mulclpi  7326  nlt1pig  7339  indpi  7340  nqnq0pi  7436  prarloclemcalc  7500
  Copyright terms: Public domain W3C validator