Theorem vprc 4114

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 4113	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2732	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 661	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  nvel  4115  intexr  4129  intnexr  4130  abnex  4425  snnex  4426  ruALT  4528  dcextest  4558  iprc  4872  snexxph  6915  elfi2  6937  fi0  6940