MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vprc Structured version   Unicode version

Theorem vprc 4333
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.
StepHypRef Expression
1 nalset 4332 . . 3  |-  -.  E. x A. y  y  e.  x
2 vex 2951 . . . . . . 7  |-  y  e. 
_V
32tbt 334 . . . . . 6  |-  ( y  e.  x  <->  ( y  e.  x  <->  y  e.  _V ) )
43albii 1575 . . . . 5  |-  ( A. y  y  e.  x  <->  A. y ( y  e.  x  <->  y  e.  _V ) )
5 dfcleq 2429 . . . . 5  |-  ( x  =  _V  <->  A. y
( y  e.  x  <->  y  e.  _V ) )
64, 5bitr4i 244 . . . 4  |-  ( A. y  y  e.  x  <->  x  =  _V )
76exbii 1592 . . 3  |-  ( E. x A. y  y  e.  x  <->  E. x  x  =  _V )
81, 7mtbi 290 . 2  |-  -.  E. x  x  =  _V
9 isset 2952 . 2  |-  ( _V  e.  _V  <->  E. x  x  =  _V )
108, 9mtbir 291 1  |-  -.  _V  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948
This theorem is referenced by:  nvel  4334  vnex  4335  intex  4348  intnex  4349  snnex  4705  iprc  5126  riotav  6546  elfi2  7411  fi0  7417  ruALT  7561  cardmin2  7877  00lsp  16049  fveqvfvv  27955  ndmaovcl  28034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-v 2950
  Copyright terms: Public domain W3C validator