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

Theorem intnexr 3879
 Description: If a class intersection is the universe, it is not a set. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intnexr ( A = V → ¬ A V)

Proof of Theorem intnexr
StepHypRef Expression
1 vprc 3862 . 2 ¬ V V
2 eleq1 2082 . 2 ( A = V → ( A V ↔ V V))
31, 2mtbiri 587 1 ( A = V → ¬ A V)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1228   ∈ wcel 1374  Vcvv 2535  ∩ cint 3589 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-ext 2004  ax-sep 3849 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-v 2537 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator