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Theorem rusbcALT 27051
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 3098 . . . 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 2998 . . . 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 2449 . . . . 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 3108 . . . . . . 7  |-  ( { x  |  x  e/  x }  e.  _V  ->  [_ { x  |  x  e/  x }  /  x ]_ x  =  { x  |  x  e/  x } )
65, 5eleq12d 2351 . . . . . 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 2449 . 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 1684   {cab 2269    e/ wnel 2447   _Vcvv 2788   [.wsbc 2991   [_csb 3081
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-nel 2449  df-v 2790  df-sbc 2992  df-csb 3082
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