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Theorem intnexr 4163
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 4147 . 2 ¬ V ∈ V
2 eleq1 2250 . 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 1363  wcel 2158  Vcvv 2749   cint 3856
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 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-v 2751
This theorem is referenced by: (None)
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