Theorem vprc 5234

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 5233	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 3435	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 322	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424

This theorem is referenced by:  nvel  5235  intex  5256  intnex  5257  abnex  7585  iprc  7734  opabn1stprc  7871  elfi2  9103  fi0  9109  ruALT  9292  cardmin2  9688  00lsp  20158  n0lplig  28746  fveqvfvv  44421  ndmaovcl  44582  vsn  46045