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Theorem ruALT 4357
Description: Alternate proof of Russell's Paradox ru 2837, simplified using (indirectly) the Axiom of Set Induction ax-setind 4343. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ruALT  |-  { x  |  x  e/  x }  e/  _V

Proof of Theorem ruALT
StepHypRef Expression
1 vprc 3963 . . 3  |-  -.  _V  e.  _V
2 df-nel 2351 . . 3  |-  ( _V 
e/  _V  <->  -.  _V  e.  _V )
31, 2mpbir 144 . 2  |-  _V  e/  _V
4 ruv 4356 . . 3  |-  { x  |  x  e/  x }  =  _V
5 neleq1 2354 . . 3  |-  ( { x  |  x  e/  x }  =  _V  ->  ( { x  |  x  e/  x }  e/  _V  <->  _V  e/  _V )
)
64, 5ax-mp 7 . 2  |-  ( { x  |  x  e/  x }  e/  _V  <->  _V  e/  _V )
73, 6mpbir 144 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074    e/ wnel 2350   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-v 2621  df-dif 2999  df-sn 3447
This theorem is referenced by: (None)
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