Theorem vprc 4265

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 4264	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2828	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 682	1 ⊢ ¬ V ∈ V

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821

This proof depends on 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 4249

This proof 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 used by:  nvel  4266  intexr  4286  intnexr  4287  abnex  4593  snnex  4594  ruALT  4698  dcextest  4728  iprc  5051  opabn1stprc  6429  snexxph  7267  elfi2  7306  fi0  7309