Theorem vprc 4260

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 4259	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2828	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 682	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4244

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823

This theorem is referenced by:  nvel  4261  intexr  4281  intnexr  4282  abnex  4588  snnex  4589  ruALT  4693  dcextest  4723  iprc  5046  opabn1stprc  6419  snexxph  7257  elfi2  7296  fi0  7299