Theorem vprc 4796

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

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nalset 4795	. . 3 |- -. E. x A. y y e. x
2		vex 3203	. . . . . . 7 |- y e. _V
3	2	tbt 359	. . . . . 6 |- ( y e. x <-> ( y e. x <-> y e. _V ) )
4	3	albii 1747	. . . . 5 |- ( A. y y e. x <-> A. y ( y e. x <-> y e. _V ) )
5		dfcleq 2616	. . . . 5 |- ( x = _V <-> A. y ( y e. x <-> y e. _V ) )
6	4, 5	bitr4i 267	. . . 4 |- ( A. y y e. x <-> x = _V )
7	6	exbii 1774	. . 3 |- ( E. x A. y y e. x <-> E. x x = _V )
8	1, 7	mtbi 312	. 2 |- -. E. x x = _V
9		isset 3207	. 2 |- ( _V e. _V <-> E. x x = _V )
10	8, 9	mtbir 313	1 |- -. _V e. _V

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  nvel  4797  vnex  4798  intex  4820  intnex  4821  abnex  6965  snnexOLD  6967  iprc  7101  opabn1stprc  7228  elfi2  8320  fi0  8326  ruALT  8508  cardmin2  8824  00lsp  18981  fveqvfvv  41204  ndmaovcl  41283