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Theorem rusbcALT 27618
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 pm5.19 351 . . 3  |-  -.  ( { x  |  x  e/  x }  e.  {
x  |  x  e/  x }  <->  -.  { x  |  x  e/  x }  e.  { x  |  x  e/  x } )
2 sbcnel12g 3270 . . . 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 3170 . . . 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 2604 . . . . 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 3280 . . . . . . 7  |-  ( { x  |  x  e/  x }  e.  _V  ->  [_ { x  |  x  e/  x }  /  x ]_ x  =  { x  |  x  e/  x } )
65, 5eleq12d 2506 . . . . . 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 287 . . . . 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 250 . . . 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 276 . . 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 170 . 2  |-  -.  {
x  |  x  e/  x }  e.  _V
11 df-nel 2604 . 2  |-  ( { x  |  x  e/  x }  e/  _V  <->  -.  { x  |  x  e/  x }  e.  _V )
1210, 11mpbir 202 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    e. wcel 1726   {cab 2424    e/ wnel 2602   _Vcvv 2958   [.wsbc 3163   [_csb 3253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-nel 2604  df-v 2960  df-sbc 3164  df-csb 3254
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