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Theorem vprc 4069
 Description: The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.)
Assertion
Ref Expression
vprc ¬ V ∈ V

Proof of Theorem vprc
StepHypRef Expression
1 vnex 4068 . 2 ¬ ∃𝑥 𝑥 = V
2 isset 2696 . 2 (V ∈ V ↔ ∃𝑥 𝑥 = V)
31, 2mtbir 661 1 ¬ V ∈ V
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2690 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 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122  ax-sep 4055 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2692 This theorem is referenced by:  nvel  4070  intexr  4084  intnexr  4085  abnex  4377  snnex  4378  ruALT  4475  dcextest  4504  iprc  4816  snexxph  6848  elfi2  6870  fi0  6873
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