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Theorem rusbcALT 27507
Description: A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (Proof modification is discouraged.)
Assertion
Ref Expression
rusbcALT  |-  { x  |  x  e/  x }  e/  _V

Proof of Theorem rusbcALT
StepHypRef Expression
1 pm5.19 350 . . 3  |-  -.  ( { x  |  x  e/  x }  e.  {
x  |  x  e/  x }  <->  -.  { x  |  x  e/  x }  e.  { x  |  x  e/  x } )
2 sbcnel12g 3228 . . . 4  |-  ( { x  |  x  e/  x }  e.  _V  ->  ( [. { x  |  x  e/  x }  /  x ]. x  e/  x  <->  [_ { x  |  x  e/  x }  /  x ]_ x  e/  [_ { x  |  x  e/  x }  /  x ]_ x ) )
3 sbc8g 3128 . . . 4  |-  ( { x  |  x  e/  x }  e.  _V  ->  ( [. { x  |  x  e/  x }  /  x ]. x  e/  x  <->  { x  |  x  e/  x }  e.  { x  |  x  e/  x } ) )
4 df-nel 2570 . . . . 5  |-  ( [_ { x  |  x  e/  x }  /  x ]_ x  e/  [_ {
x  |  x  e/  x }  /  x ]_ x  <->  -.  [_ { x  |  x  e/  x }  /  x ]_ x  e.  [_ { x  |  x  e/  x }  /  x ]_ x )
5 csbvarg 3238 . . . . . . 7  |-  ( { x  |  x  e/  x }  e.  _V  ->  [_ { x  |  x  e/  x }  /  x ]_ x  =  { x  |  x  e/  x } )
65, 5eleq12d 2472 . . . . . 6  |-  ( { x  |  x  e/  x }  e.  _V  ->  ( [_ { x  |  x  e/  x }  /  x ]_ x  e.  [_ { x  |  x  e/  x }  /  x ]_ x  <->  { x  |  x  e/  x }  e.  { x  |  x  e/  x } ) )
76notbid 286 . . . . 5  |-  ( { x  |  x  e/  x }  e.  _V  ->  ( -.  [_ {
x  |  x  e/  x }  /  x ]_ x  e.  [_ {
x  |  x  e/  x }  /  x ]_ x  <->  -.  { x  |  x  e/  x }  e.  { x  |  x  e/  x } ) )
84, 7syl5bb 249 . . . 4  |-  ( { x  |  x  e/  x }  e.  _V  ->  ( [_ { x  |  x  e/  x }  /  x ]_ x  e/  [_ { x  |  x  e/  x }  /  x ]_ x  <->  -.  { x  |  x  e/  x }  e.  { x  |  x  e/  x } ) )
92, 3, 83bitr3d 275 . . 3  |-  ( { x  |  x  e/  x }  e.  _V  ->  ( { x  |  x  e/  x }  e.  { x  |  x  e/  x }  <->  -.  { x  |  x  e/  x }  e.  { x  |  x  e/  x } ) )
101, 9mto 169 . 2  |-  -.  {
x  |  x  e/  x }  e.  _V
11 df-nel 2570 . 2  |-  ( { x  |  x  e/  x }  e/  _V  <->  -.  { x  |  x  e/  x }  e.  _V )
1210, 11mpbir 201 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    e. wcel 1721   {cab 2390    e/ wnel 2568   _Vcvv 2916   [.wsbc 3121   [_csb 3211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-nel 2570  df-v 2918  df-sbc 3122  df-csb 3212
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