Theorem vprc 5257

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

Step	Hyp	Ref	Expression
1		vnex 5256	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 3452	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 323	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440

This theorem is referenced by:  nvel  5258  intex  5286  intnex  5287  abnex  7697  iprc  7851  opabn1stprc  8000  elfi2  9323  fi0  9329  ruALT  9517  cardmin2  9914  00lsp  20903  n0lplig  30446  fveqvfvv  47044  ndmaovcl  47207  vsn  48816  posnex  48984  prsnex  48985