Theorem vprc 5256

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 5255	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 3454	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 322	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1540  ∃wex 1780   ∈ wcel 2105  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5240

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443

This theorem is referenced by:  nvel  5257  intex  5278  intnex  5279  abnex  7661  iprc  7820  opabn1stprc  7958  elfi2  9263  fi0  9269  ruALT  9452  cardmin2  9848  00lsp  20341  n0lplig  29074  fveqvfvv  44874  ndmaovcl  45035  vsn  46497