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Theorem ruALT 4302
Description: Alternate proof of Russell's Paradox ru 2815, simplified using (indirectly) the Axiom of Set Induction ax-setind 4288. (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 3917 . . 3  |-  -.  _V  e.  _V
2 df-nel 2341 . . 3  |-  ( _V 
e/  _V  <->  -.  _V  e.  _V )
31, 2mpbir 144 . 2  |-  _V  e/  _V
4 ruv 4301 . . 3  |-  { x  |  x  e/  x }  =  _V
5 neleq1 2344 . . 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 1285    e. wcel 1434   {cab 2068    e/ wnel 2340   _Vcvv 2602
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-v 2604  df-dif 2976  df-sn 3412
This theorem is referenced by: (None)
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