Theorem vprc 4215

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 4214	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2806	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 675	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799

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 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801

This theorem is referenced by:  nvel  4216  intexr  4233  intnexr  4234  abnex  4537  snnex  4538  ruALT  4642  dcextest  4672  iprc  4992  snexxph  7113  elfi2  7135  fi0  7138