Theorem vprc 4147

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 4146	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2755	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 672	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1363  ∃wex 1502   ∈ wcel 2158  Vcvv 2749

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 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-v 2751

This theorem is referenced by:  nvel  4148  intexr  4162  intnexr  4163  abnex  4459  snnex  4460  ruALT  4562  dcextest  4592  iprc  4907  snexxph  6962  elfi2  6984  fi0  6987