Theorem vprc 5221

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 5220	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 3508	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 325	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3496

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2795  ax-sep 5205

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-v 3498

This theorem is referenced by:  nvel  5222  intex  5242  intnex  5243  abnex  7481  iprc  7620  opabn1stprc  7758  elfi2  8880  fi0  8886  ruALT  9069  cardmin2  9429  00lsp  19755  n0lplig  28262  fveqvfvv  43282  ndmaovcl  43409