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Theorem ruALT 4466
Description: Alternate proof of Russell's Paradox ru 2908, simplified using (indirectly) the Axiom of Set Induction ax-setind 4452. (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 4060 . . 3  |-  -.  _V  e.  _V
2 df-nel 2404 . . 3  |-  ( _V 
e/  _V  <->  -.  _V  e.  _V )
31, 2mpbir 145 . 2  |-  _V  e/  _V
4 ruv 4465 . . 3  |-  { x  |  x  e/  x }  =  _V
5 neleq1 2407 . . 3  |-  ( { x  |  x  e/  x }  =  _V  ->  ( { x  |  x  e/  x }  e/  _V  <->  _V  e/  _V )
)
64, 5ax-mp 5 . 2  |-  ( { x  |  x  e/  x }  e/  _V  <->  _V  e/  _V )
73, 6mpbir 145 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 104    = wceq 1331    e. wcel 1480   {cab 2125    e/ wnel 2403   _Vcvv 2686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-v 2688  df-dif 3073  df-sn 3533
This theorem is referenced by: (None)
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