Theorem vprc 5273

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 5272	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 3464	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 323	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452

This theorem is referenced by:  nvel  5274  intex  5302  intnex  5303  abnex  7736  iprc  7890  opabn1stprc  8040  elfi2  9372  fi0  9378  ruALT  9563  cardmin2  9959  00lsp  20894  n0lplig  30419  fveqvfvv  47045  ndmaovcl  47208  vsn  48804  posnex  48972  prsnex  48973