Theorem vprc 4184

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 4183	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2780	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 673	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-v 2775

This theorem is referenced by:  nvel  4185  intexr  4202  intnexr  4203  abnex  4502  snnex  4503  ruALT  4607  dcextest  4637  iprc  4956  snexxph  7067  elfi2  7089  fi0  7092