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

Theorem intnexr 4166
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 4150 . 2 ¬ V ∈ V
2 eleq1 2252 . 2 ( 𝐴 = V → ( 𝐴 ∈ V ↔ V ∈ V))
31, 2mtbiri 676 1 ( 𝐴 = V → ¬ 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1364  wcel 2160  Vcvv 2752   cint 3859
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 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator