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Theorem ru 3120
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 4307 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 4299, Pairing prex 4366, Union uniex 4664, Power Set pwex 4342, and Infinity omex 7554 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 5490 (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 8311 and Cantor's Theorem canth 6498 are provably false! (See ncanth 6499 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 4290 replaces ax-rep 4280) with ax-sep 4290 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 7521 (derived from the Axiom of Regularity), so for us the Russell class equals the universe 
_V (theorem ruv 7524). See ruALT 7525 for an alternate proof of ru 3120 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 pm5.19 350 . . . . . 6  |-  -.  (
y  e.  y  <->  -.  y  e.  y )
2 eleq1 2464 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  y  <->  y  e.  y ) )
3 df-nel 2570 . . . . . . . . 9  |-  ( x  e/  x  <->  -.  x  e.  x )
4 id 20 . . . . . . . . . . 11  |-  ( x  =  y  ->  x  =  y )
54, 4eleq12d 2472 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  e.  x  <->  y  e.  y ) )
65notbid 286 . . . . . . . . 9  |-  ( x  =  y  ->  ( -.  x  e.  x  <->  -.  y  e.  y ) )
73, 6syl5bb 249 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e/  x  <->  -.  y  e.  y ) )
82, 7bibi12d 313 . . . . . . 7  |-  ( x  =  y  ->  (
( x  e.  y  <-> 
x  e/  x )  <->  ( y  e.  y  <->  -.  y  e.  y ) ) )
98spv 1963 . . . . . 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 2509 . . . . 5  |-  ( y  =  { x  |  x  e/  x }  <->  A. x ( x  e.  y  <->  x  e/  x
) )
1210, 11mtbir 291 . . . 4  |-  -.  y  =  { x  |  x  e/  x }
1312nex 1561 . . 3  |-  -.  E. y  y  =  {
x  |  x  e/  x }
14 isset 2920 . . 3  |-  ( { x  |  x  e/  x }  e.  _V  <->  E. y  y  =  {
x  |  x  e/  x } )
1513, 14mtbir 291 . 2  |-  -.  {
x  |  x  e/  x }  e.  _V
16 df-nel 2570 . 2  |-  ( { x  |  x  e/  x }  e/  _V  <->  -.  { x  |  x  e/  x }  e.  _V )
1715, 16mpbir 201 1  |-  { x  |  x  e/  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390    e/ wnel 2568   _Vcvv 2916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nel 2570  df-v 2918
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