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

Theorem ruALT 4229
Description: Alternate proof of Russell's Paradox ru 2757, simplified using (indirectly) the Axiom of Set Induction ax-setind 4220. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ruALT {xxx} ∉ V

Proof of Theorem ruALT
StepHypRef Expression
1 vprc 3879 . . 3 ¬ V V
2 df-nel 2204 . . 3 (V ∉ V ↔ ¬ V V)
31, 2mpbir 134 . 2 V ∉ V
4 ruv 4228 . . 3 {xxx} = V
5 neleq1 2295 . . 3 ({xxx} = V → ({xxx} ∉ V ↔ V ∉ V))
64, 5ax-mp 7 . 2 ({xxx} ∉ V ↔ V ∉ V)
73, 6mpbir 134 1 {xxx} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98   = wceq 1242   wcel 1390  {cab 2023  wnel 2202  Vcvv 2551
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-v 2553  df-dif 2914  df-sn 3373
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator