Theorem vprc 3888

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 3887	. . 3 |- -. E. x A. y y e. x
2		vex 2560	. . . . . . 7 |- y e. _V
3	2	tbt 236	. . . . . 6 |- ( y e. x <-> ( y e. x <-> y e. _V ) )
4	3	albii 1359	. . . . 5 |- ( A. y y e. x <-> A. y ( y e. x <-> y e. _V ) )
5		dfcleq 2034	. . . . 5 |- ( x = _V <-> A. y ( y e. x <-> y e. _V ) )
6	4, 5	bitr4i 176	. . . 4 |- ( A. y y e. x <-> x = _V )
7	6	exbii 1496	. . 3 |- ( E. x A. y y e. x <-> E. x x = _V )
8	1, 7	mtbi 595	. 2 |- -. E. x x = _V
9		isset 2561	. 2 |- ( _V e. _V <-> E. x x = _V )
10	8, 9	mtbir 596	1 |- -. _V e. _V

Syntax hints:   -. wn 3   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022  ax-sep 3875

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by:  nvel  3889  vnex  3890  intexr  3904  intnexr  3905  snnex  4181  ruALT  4275  iprc  4600