ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  vprc Structured version   GIF version

Theorem vprc 3861
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 V

Proof of Theorem vprc
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nalset 3860 . . 3 ¬ xy y x
2 vex 2537 . . . . . . 7 y V
32tbt 236 . . . . . 6 (y x ↔ (y xy V))
43albii 1339 . . . . 5 (y y xy(y xy V))
5 dfcleq 2017 . . . . 5 (x = V ↔ y(y xy V))
64, 5bitr4i 176 . . . 4 (y y xx = V)
76exbii 1480 . . 3 (xy y xx x = V)
81, 7mtbi 582 . 2 ¬ x x = V
9 isset 2538 . 2 (V V ↔ x x = V)
108, 9mtbir 583 1 ¬ V V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98  wal 1226   = wceq 1228  wex 1363   wcel 1375  Vcvv 2534
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 532  ax-in2 533  ax-5 1316  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-ext 2005  ax-sep 3848
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-v 2536
This theorem is referenced by:  nvel  3862  vnex  3863  intexr  3877  intnexr  3878  snnex  4129  ruALT  4211  iprc  4525
  Copyright terms: Public domain W3C validator