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Theorem bj-ru1 31538
Description: A version of Russell's paradox ru 3266 (see also bj-ru 31539). Note the more economical use of bj-abeq2 31388 instead of abeq2 2560 to avoid dependency on ax-13 2091. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ru1  |-  -.  E. y  y  =  {
x  |  -.  x  e.  x }
Distinct variable group:    x, y

Proof of Theorem bj-ru1
StepHypRef Expression
1 bj-ru0 31537 . . 3  |-  -.  A. x ( x  e.  y  <->  -.  x  e.  x )
2 bj-abeq2 31388 . . 3  |-  ( y  =  { x  |  -.  x  e.  x } 
<-> 
A. x ( x  e.  y  <->  -.  x  e.  x ) )
31, 2mtbir 301 . 2  |-  -.  y  =  { x  |  -.  x  e.  x }
43nex 1678 1  |-  -.  E. y  y  =  {
x  |  -.  x  e.  x }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188   A.wal 1442    = wceq 1444   E.wex 1663   {cab 2437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447
This theorem is referenced by:  bj-ru  31539
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