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Theorem vprc 3913
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
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nalset 3912 . . 3 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 2575 . . . . . . 7 𝑦 ∈ V
32tbt 240 . . . . . 6 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1373 . . . . 5 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2048 . . . . 5 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 180 . . . 4 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1510 . . 3 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 603 . 2 ¬ ∃𝑥 𝑥 = V
9 isset 2576 . 2 (V ∈ V ↔ ∃𝑥 𝑥 = V)
108, 9mtbir 604 1 ¬ V ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 102  wal 1255   = wceq 1257  wex 1395  wcel 1407  Vcvv 2572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-5 1350  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-ext 2036  ax-sep 3900
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-v 2574
This theorem is referenced by:  nvel  3914  vnex  3915  intexr  3929  intnexr  3930  snnex  4206  ruALT  4300  iprc  4625
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