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

Theorem vprc 4020
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
StepHypRef Expression
1 vnex 4019 . 2 ¬ ∃𝑥 𝑥 = V
2 isset 2663 . 2 (V ∈ V ↔ ∃𝑥 𝑥 = V)
31, 2mtbir 643 1 ¬ V ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1314  wex 1451  wcel 1463  Vcvv 2657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097  ax-sep 4006
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2659
This theorem is referenced by:  nvel  4021  intexr  4035  intnexr  4036  abnex  4328  snnex  4329  ruALT  4426  dcextest  4455  iprc  4765  snexxph  6790  elfi2  6812  fi0  6815
  Copyright terms: Public domain W3C validator