Theorem vprc 3909

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 3908	. . 3 |- -. E. x A. y y e. x
2		vex 2604	. . . . . . 7 |- y e. _V
3	2	tbt 245	. . . . . 6 |- ( y e. x <-> ( y e. x <-> y e. _V ) )
4	3	albii 1399	. . . . 5 |- ( A. y y e. x <-> A. y ( y e. x <-> y e. _V ) )
5		dfcleq 2075	. . . . 5 |- ( x = _V <-> A. y ( y e. x <-> y e. _V ) )
6	4, 5	bitr4i 185	. . . 4 |- ( A. y y e. x <-> x = _V )
7	6	exbii 1536	. . 3 |- ( E. x A. y y e. x <-> E. x x = _V )
8	1, 7	mtbi 627	. 2 |- -. E. x x = _V
9		isset 2605	. 2 |- ( _V e. _V <-> E. x x = _V )
10	8, 9	mtbir 628	1 |- -. _V e. _V

Syntax hints:   -. wn 3   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601

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 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  nvel  3910  vnex  3911  intexr  3925  intnexr  3926  snnex  4199  ruALT  4294  iprc  4618