Theorem vprc 4175

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 4174	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2777	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 672	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1372  ∃wex 1514   ∈ wcel 2175  Vcvv 2771

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 615  ax-in2 616  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-v 2773

This theorem is referenced by:  nvel  4176  intexr  4193  intnexr  4194  abnex  4492  snnex  4493  ruALT  4597  dcextest  4627  iprc  4944  snexxph  7034  elfi2  7056  fi0  7059