Theorem vprc 4161

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 4160	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2766	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 672	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1364  ∃wex 1503   ∈ wcel 2164  Vcvv 2760

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762

This theorem is referenced by:  nvel  4162  intexr  4179  intnexr  4180  abnex  4478  snnex  4479  ruALT  4583  dcextest  4613  iprc  4930  snexxph  7009  elfi2  7031  fi0  7034