Theorem vprc 4165

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 4164	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2769	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 672	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by:  nvel  4166  intexr  4183  intnexr  4184  abnex  4482  snnex  4483  ruALT  4587  dcextest  4617  iprc  4934  snexxph  7016  elfi2  7038  fi0  7041