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

Theorem dtru 4475
 Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4474. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
dtru ¬ ∀𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem dtru
StepHypRef Expression
1 dtruex 4474 . 2 𝑥 ¬ 𝑥 = 𝑦
2 exnalim 1625 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
31, 2ax-mp 5 1 ¬ ∀𝑥 𝑥 = 𝑦
 Colors of variables: wff set class Syntax hints:  ¬ wn 3  ∀wal 1329  ∃wex 1468 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-setind 4452 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533 This theorem is referenced by:  oprabidlem  5802
 Copyright terms: Public domain W3C validator