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Theorem ru 3003
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 4174 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 4166, Pairing prex 4233, Union uniex 4532, Power Set pwex 4209, and Infinity omex 7360 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 5346 (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 8118 and Cantor's Theorem canth 6310 are provably false! (See ncanth 6311 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 4157 replaces ax-rep 4147) with ax-sep 4157 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 7327 (derived from the Axiom of Regularity), so for us the Russell class equals the universe 
_V (theorem ruv 7330). See ruALT 7331 for an alternate proof of ru 3003 derived from that fact. (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 pm5.19 349 . . . . . 6  |-  -.  (
y  e.  y  <->  -.  y  e.  y )
2 eleq1 2356 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  y  <->  y  e.  y ) )
3 df-nel 2462 . . . . . . . . 9  |-  ( x  e/  x  <->  -.  x  e.  x )
4 id 19 . . . . . . . . . . 11  |-  ( x  =  y  ->  x  =  y )
54, 4eleq12d 2364 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  e.  x  <->  y  e.  y ) )
65notbid 285 . . . . . . . . 9  |-  ( x  =  y  ->  ( -.  x  e.  x  <->  -.  y  e.  y ) )
73, 6syl5bb 248 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e/  x  <->  -.  y  e.  y ) )
82, 7bibi12d 312 . . . . . . 7  |-  ( x  =  y  ->  (
( x  e.  y  <-> 
x  e/  x )  <->  ( y  e.  y  <->  -.  y  e.  y ) ) )
98spv 1951 . . . . . 6  |-  ( A. x ( x  e.  y  <->  x  e/  x
)  ->  ( y  e.  y  <->  -.  y  e.  y ) )
101, 9mto 167 . . . . 5  |-  -.  A. x ( x  e.  y  <->  x  e/  x
)
11 abeq2 2401 . . . . 5  |-  ( y  =  { x  |  x  e/  x }  <->  A. x ( x  e.  y  <->  x  e/  x
) )
1210, 11mtbir 290 . . . 4  |-  -.  y  =  { x  |  x  e/  x }
1312nex 1545 . . 3  |-  -.  E. y  y  =  {
x  |  x  e/  x }
14 isset 2805 . . 3  |-  ( { x  |  x  e/  x }  e.  _V  <->  E. y  y  =  {
x  |  x  e/  x } )
1513, 14mtbir 290 . 2  |-  -.  {
x  |  x  e/  x }  e.  _V
16 df-nel 2462 . 2  |-  ( { x  |  x  e/  x }  e/  _V  <->  -.  { x  |  x  e/  x }  e.  _V )
1715, 16mpbir 200 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    e/ wnel 2460   _Vcvv 2801
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nel 2462  df-v 2803
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