Theorem vprc 3963

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 e. _V

Proof of Theorem vprc

Step	Hyp	Ref	Expression
1		vnex 3962	. 2 |- -. E. x x = _V
2		isset 2625	. 2 |- ( _V e. _V <-> E. x x = _V )
3	1, 2	mtbir 631	1 |- -. _V e. _V

Syntax hints:   -. wn 3   = wceq 1289   E.wex 1426   e. wcel 1438   _Vcvv 2619

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070  ax-sep 3949

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621

This theorem is referenced by:  nvel  3964  intexr  3978  intnexr  3979  snnex  4262  ruALT  4357  dcextest  4386  iprc  4689  snexxph  6638