Theorem vprc 5211

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 5210	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 3506	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 325	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793  ax-sep 5195

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3496

This theorem is referenced by:  nvel  5212  intex  5232  intnex  5233  abnex  7473  iprc  7612  opabn1stprc  7750  elfi2  8872  fi0  8878  ruALT  9061  cardmin2  9421  00lsp  19747  n0lplig  28254  fveqvfvv  43269  ndmaovcl  43396