Theorem vprc 4242

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 4241	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2820	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 678	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2813

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 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-v 2815

This theorem is referenced by:  nvel  4243  intexr  4262  intnexr  4263  abnex  4568  snnex  4569  ruALT  4673  dcextest  4703  iprc  5026  opabn1stprc  6389  snexxph  7220  elfi2  7259  fi0  7262