Theorem vprc 4247

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 4246	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2822	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 678	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1398  ∃wex 1541   ∈ wcel 2205  Vcvv 2815

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-v 2817

This theorem is referenced by:  nvel  4248  intexr  4267  intnexr  4268  abnex  4573  snnex  4574  ruALT  4678  dcextest  4708  iprc  5031  opabn1stprc  6402  snexxph  7233  elfi2  7272  fi0  7275