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Theorem ruALT 7569
Description: Alternate proof of Russell's Paradox ru 3160, simplified using (indirectly) the Axiom of Regularity ax-reg 7560. (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 4341 . . 3  |-  -.  _V  e.  _V
2 df-nel 2602 . . 3  |-  ( _V 
e/  _V  <->  -.  _V  e.  _V )
31, 2mpbir 201 . 2  |-  _V  e/  _V
4 ruv 7568 . . 3  |-  { x  |  x  e/  x }  =  _V
5 neleq1 2699 . . 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 201 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2422    e/ wnel 2600   _Vcvv 2956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-reg 7560
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-un 3325  df-nul 3629  df-sn 3820  df-pr 3821
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