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Theorem ru 2737
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 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 {xxx} (the "Russell class") for A, it asserted {xxx} V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {xxx} ∉ 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 3822. (Contributed by NM, 7-Aug-1994.)

Assertion
Ref Expression
ru {xxx} ∉ V

Proof of Theorem ru
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 pm5.19 609 . . . . . 6 ¬ (y y ↔ ¬ y y)
2 eleq1 2081 . . . . . . . 8 (x = y → (x yy y))
3 df-nel 2188 . . . . . . . . 9 (xx ↔ ¬ x x)
4 id 19 . . . . . . . . . . 11 (x = yx = y)
54, 4eleq12d 2089 . . . . . . . . . 10 (x = y → (x xy y))
65notbid 579 . . . . . . . . 9 (x = y → (¬ x x ↔ ¬ y y))
73, 6syl5bb 181 . . . . . . . 8 (x = y → (xx ↔ ¬ y y))
82, 7bibi12d 224 . . . . . . 7 (x = y → ((x yxx) ↔ (y y ↔ ¬ y y)))
98spv 1723 . . . . . 6 (x(x yxx) → (y y ↔ ¬ y y))
101, 9mto 575 . . . . 5 ¬ x(x yxx)
11 abeq2 2127 . . . . 5 (y = {xxx} ↔ x(x yxx))
1210, 11mtbir 583 . . . 4 ¬ y = {xxx}
1312nex 1372 . . 3 ¬ y y = {xxx}
14 isset 2536 . . 3 ({xxx} V ↔ y y = {xxx})
1513, 14mtbir 583 . 2 ¬ {xxx} V
16 df-nel 2188 . 2 ({xxx} ∉ V ↔ ¬ {xxx} V)
1715, 16mpbir 134 1 {xxx} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98  wal 1228   = wceq 1230  wex 1364   wcel 1376  {cab 2007  wnel 2186  Vcvv 2532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-5 1318  ax-7 1319  ax-gen 1320  ax-ie1 1365  ax-ie2 1366  ax-8 1378  ax-11 1380  ax-4 1383  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2003
This theorem depends on definitions:  df-bi 110  df-tru 1233  df-fal 1236  df-nf 1332  df-sb 1629  df-clab 2008  df-cleq 2014  df-clel 2017  df-nel 2188  df-v 2534
This theorem is referenced by: (None)
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