Theorem vprc 4244

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 4243	. 2 |- -. E. x x = _V
2		isset 2822	. 2 |- ( _V e. _V <-> E. x x = _V )
3	1, 2	mtbir 678	1 |- -. _V e. _V

Syntax hints:   -. wn 3   = wceq 1398   E.wex 1541   e. wcel 2205   _Vcvv 2815

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 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-v 2817

This theorem is referenced by:  nvel  4245  intexr  4264  intnexr  4265  abnex  4570  snnex  4571  ruALT  4675  dcextest  4705  iprc  5028  opabn1stprc  6391  snexxph  7222  elfi2  7261  fi0  7264