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

Theorem ruALT 4213
 Description: Alternate proof of Russell's Paradox ru 2740, simplified using (indirectly) the Axiom of Set Induction ax-setind 4204. (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 3862 . . 3 ¬ V V
2 df-nel 2189 . . 3 (V ∉ V ↔ ¬ V V)
31, 2mpbir 134 . 2 V ∉ V
4 ruv 4212 . . 3 {xxx} = V
5 neleq1 2279 . . 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 1228   ∈ wcel 1374  {cab 2008   ∉ wnel 2187  Vcvv 2535 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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-nel 2189  df-ral 2289  df-v 2537  df-dif 2897  df-sn 3356 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator