Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  neq0r GIF version

Theorem neq0r 3269
 Description: An inhabited class is nonempty. See n0rf 3267 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
Assertion
Ref Expression
neq0r (∃𝑥 𝑥𝐴 → ¬ 𝐴 = ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem neq0r
StepHypRef Expression
1 n0r 3268 . 2 (∃𝑥 𝑥𝐴𝐴 ≠ ∅)
21neneqd 2267 1 (∃𝑥 𝑥𝐴 → ¬ 𝐴 = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1285  ∃wex 1422   ∈ wcel 1434  ∅c0 3258 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-v 2604  df-dif 2976  df-nul 3259 This theorem is referenced by:  fzn  9137
 Copyright terms: Public domain W3C validator