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Theorem ru 2786
Description: Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.

In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as 
A  e.  _V, asserted that any collection of sets  A is a set i.e. belongs to the universe 
_V of all sets. In particular, by substituting  { x  |  x  e/  x } (the "Russell class") for  A, it asserted  { x  |  x  e/  x }  e.  _V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove  { x  |  x  e/  x }  e/  _V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system.

In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom asserting that  A is a set only when it is smaller than some other set  B. The intuitionistic set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla] p. "Axioms of CZF and IZF", which we include as ax-sep 3903. (Contributed by NM, 7-Aug-1994.)

Assertion
Ref Expression
ru  |-  { x  |  x  e/  x }  e/  _V

Proof of Theorem ru
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 pm5.19 632 . . . . . 6  |-  -.  (
y  e.  y  <->  -.  y  e.  y )
2 eleq1 2116 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  y  <->  y  e.  y ) )
3 df-nel 2315 . . . . . . . . 9  |-  ( x  e/  x  <->  -.  x  e.  x )
4 id 19 . . . . . . . . . . 11  |-  ( x  =  y  ->  x  =  y )
54, 4eleq12d 2124 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  e.  x  <->  y  e.  y ) )
65notbid 602 . . . . . . . . 9  |-  ( x  =  y  ->  ( -.  x  e.  x  <->  -.  y  e.  y ) )
73, 6syl5bb 185 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e/  x  <->  -.  y  e.  y ) )
82, 7bibi12d 228 . . . . . . 7  |-  ( x  =  y  ->  (
( x  e.  y  <-> 
x  e/  x )  <->  ( y  e.  y  <->  -.  y  e.  y ) ) )
98spv 1756 . . . . . 6  |-  ( A. x ( x  e.  y  <->  x  e/  x
)  ->  ( y  e.  y  <->  -.  y  e.  y ) )
101, 9mto 598 . . . . 5  |-  -.  A. x ( x  e.  y  <->  x  e/  x
)
11 abeq2 2162 . . . . 5  |-  ( y  =  { x  |  x  e/  x }  <->  A. x ( x  e.  y  <->  x  e/  x
) )
1210, 11mtbir 606 . . . 4  |-  -.  y  =  { x  |  x  e/  x }
1312nex 1405 . . 3  |-  -.  E. y  y  =  {
x  |  x  e/  x }
14 isset 2578 . . 3  |-  ( { x  |  x  e/  x }  e.  _V  <->  E. y  y  =  {
x  |  x  e/  x } )
1513, 14mtbir 606 . 2  |-  -.  {
x  |  x  e/  x }  e.  _V
16 df-nel 2315 . 2  |-  ( { x  |  x  e/  x }  e/  _V  <->  -.  { x  |  x  e/  x }  e.  _V )
1715, 16mpbir 138 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042    e/ wnel 2314   _Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nel 2315  df-v 2576
This theorem is referenced by: (None)
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