Theorem vprc 5320

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 5319	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 3491	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 323	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1536  ∃wex 1775   ∈ wcel 2105  Vcvv 3477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479

This theorem is referenced by:  nvel  5321  intex  5349  intnex  5350  abnex  7775  iprc  7933  opabn1stprc  8081  elfi2  9451  fi0  9457  ruALT  9640  cardmin2  10036  00lsp  20996  n0lplig  30511  fveqvfvv  46989  ndmaovcl  47152  vsn  48659