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Theorem rusbcALT 27742
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 pm5.19 349 . . 3  |-  -.  ( { x  |  x  e/  x }  e.  {
x  |  x  e/  x }  <->  -.  { x  |  x  e/  x }  e.  { x  |  x  e/  x } )
2 sbcnel12g 3111 . . . 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 3011 . . . 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 2462 . . . . 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 3121 . . . . . . 7  |-  ( { x  |  x  e/  x }  e.  _V  ->  [_ { x  |  x  e/  x }  /  x ]_ x  =  { x  |  x  e/  x } )
65, 5eleq12d 2364 . . . . . 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 285 . . . . 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 248 . . . 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 274 . . 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 167 . 2  |-  -.  {
x  |  x  e/  x }  e.  _V
11 df-nel 2462 . 2  |-  ( { x  |  x  e/  x }  e/  _V  <->  -.  { x  |  x  e/  x }  e.  _V )
1210, 11mpbir 200 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    e. wcel 1696   {cab 2282    e/ wnel 2460   _Vcvv 2801   [.wsbc 3004   [_csb 3094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-nel 2462  df-v 2803  df-sbc 3005  df-csb 3095
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