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Theorem ruALT 7315
Description: Alternate proof of Russell's Paradox ru 2990, simplified using (indirectly) the Axiom of Regularity ax-reg 7306. (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 4152 . . 3  |-  -.  _V  e.  _V
2 df-nel 2449 . . 3  |-  ( _V 
e/  _V  <->  -.  _V  e.  _V )
31, 2mpbir 200 . 2  |-  _V  e/  _V
4 ruv 7314 . . 3  |-  { x  |  x  e/  x }  =  _V
5 neleq1 2537 . . 3  |-  ( { x  |  x  e/  x }  =  _V  ->  ( { x  |  x  e/  x }  e/  _V  <->  _V  e/  _V )
)
64, 5ax-mp 8 . 2  |-  ( { x  |  x  e/  x }  e/  _V  <->  _V  e/  _V )
73, 6mpbir 200 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269    e/ wnel 2447   _Vcvv 2788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647
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