Theorem vprc 4150

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 4149	. 2 |- -. E. x x = _V
2		isset 2758	. 2 |- ( _V e. _V <-> E. x x = _V )
3	1, 2	mtbir 672	1 |- -. _V e. _V

Syntax hints:   -. wn 3   = wceq 1364   E.wex 1503   e. wcel 2160   _Vcvv 2752

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754

This theorem is referenced by:  nvel  4151  intexr  4165  intnexr  4166  abnex  4462  snnex  4463  ruALT  4565  dcextest  4595  iprc  4910  snexxph  6974  elfi2  6996  fi0  6999