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Theorem intnexr 4044
 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 4028 . 2 ¬ V ∈ V
2 eleq1 2178 . 2 ( 𝐴 = V → ( 𝐴 ∈ V ↔ V ∈ V))
31, 2mtbiri 647 1 ( 𝐴 = V → ¬ 𝐴 ∈ V)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1314   ∈ wcel 1463  Vcvv 2658  ∩ cint 3739 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 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097  ax-sep 4014 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660 This theorem is referenced by: (None)
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