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Theorem ru 2920
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 ssex 4055 asserting that  A is a set only when it is smaller than some other set  B. However, Zermelo was then faced with a "chicken and egg" problem of how to show  B is a set, leading him to introduce the set-building axioms of Null Set 0ex 4047, Pairing prex 4111, Union uniex 4407, Power Set pwex 4087, and Infinity omex 7228 to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 5187 (whose modern formalization is due to Skolem, also in 1922). Thus in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics!

Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than set variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287).

Russell himself continued in a different direction, avoiding the paradox with his "theory of types." Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contradicting ZF and NBG set theories, and it has other bizarre consequences: when sets become too huge (beyond the size of those used in standard mathematics), the Axiom of Choice ac4 7986 and Cantor's Theorem canth 6178 are provably false! (See ncanth 6179 for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep 4038 replaces ax-rep 4028) with ax-sep 4038 restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic," J. Symb. Logic 9:1-19 (1944).

Under our ZF set theory, every set is a member of the Russell class by elirrv 7195 (derived from the Axiom of Regularity), so for us the Russell class equals the universe 
_V (theorem ruv 7198). See ruALT 7199 for an alternate proof of ru 2920 derived from that fact. (Contributed by NM, 7-Aug-1994.)

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

Proof of Theorem ru
StepHypRef Expression
1 pm5.19 351 . . . . . 6  |-  -.  (
y  e.  y  <->  -.  y  e.  y )
2 eleq1 2313 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  y  <->  y  e.  y ) )
3 df-nel 2415 . . . . . . . . 9  |-  ( x  e/  x  <->  -.  x  e.  x )
4 id 21 . . . . . . . . . . 11  |-  ( x  =  y  ->  x  =  y )
54, 4eleq12d 2321 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  e.  x  <->  y  e.  y ) )
65notbid 287 . . . . . . . . 9  |-  ( x  =  y  ->  ( -.  x  e.  x  <->  -.  y  e.  y ) )
73, 6syl5bb 250 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e/  x  <->  -.  y  e.  y ) )
82, 7bibi12d 314 . . . . . . 7  |-  ( x  =  y  ->  (
( x  e.  y  <-> 
x  e/  x )  <->  ( y  e.  y  <->  -.  y  e.  y ) ) )
98a4v 1996 . . . . . 6  |-  ( A. x ( x  e.  y  <->  x  e/  x
)  ->  ( y  e.  y  <->  -.  y  e.  y ) )
101, 9mto 169 . . . . 5  |-  -.  A. x ( x  e.  y  <->  x  e/  x
)
11 abeq2 2354 . . . . 5  |-  ( y  =  { x  |  x  e/  x }  <->  A. x ( x  e.  y  <->  x  e/  x
) )
1210, 11mtbir 292 . . . 4  |-  -.  y  =  { x  |  x  e/  x }
1312nex 1587 . . 3  |-  -.  E. y  y  =  {
x  |  x  e/  x }
14 isset 2731 . . 3  |-  ( { x  |  x  e/  x }  e.  _V  <->  E. y  y  =  {
x  |  x  e/  x } )
1513, 14mtbir 292 . 2  |-  -.  {
x  |  x  e/  x }  e.  _V
16 df-nel 2415 . 2  |-  ( { x  |  x  e/  x }  e/  _V  <->  -.  { x  |  x  e/  x }  e.  _V )
1715, 16mpbir 202 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2239    e/ wnel 2413   _Vcvv 2727
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nel 2415  df-v 2729
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