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Theorem ru 2643
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". (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 pm5.19 604 . . . . . 6 ¬ (y y ↔ ¬ y y)
2 eleq1 2005 . . . . . . . 8 (x = y → (x yy y))
3 df-nel 2110 . . . . . . . . 9 (xx ↔ ¬ x x)
4 id 17 . . . . . . . . . . 11 (x = yx = y)
54, 4eleq12d 2013 . . . . . . . . . 10 (x = y → (x xy y))
65notbid 574 . . . . . . . . 9 (x = y → (¬ x x ↔ ¬ y y))
73, 6syl5bb 179 . . . . . . . 8 (x = y → (xx ↔ ¬ y y))
82, 7bibi12d 222 . . . . . . 7 (x = y → ((x yxx) ↔ (y y ↔ ¬ y y)))
98spv 1679 . . . . . 6 (x(x yxx) → (y y ↔ ¬ y y))
101, 9mto 570 . . . . 5 ¬ x(x yxx)
11 abeq2 2050 . . . . 5 (y = {xxx} ↔ x(x yxx))
1210, 11mtbir 578 . . . 4 ¬ y = {xxx}
1312nex 1336 . . 3 ¬ y y = {xxx}
14 isset 2445 . . 3 ({xxx} V ↔ y y = {xxx})
1513, 14mtbir 578 . 2 ¬ {xxx} V
16 df-nel 2110 . 2 ({xxx} ∉ V ↔ ¬ {xxx} V)
1715, 16mpbir 132 1 {xxx} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 96  wal 1281  wex 1328   = wceq 1340   wcel 1342  {cab 1931  wnel 2108  Vcvv 2441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-in1 528  ax-in2 529  ax-5 1282  ax-7 1283  ax-gen 1284  ax-ie1 1329  ax-ie2 1330  ax-8 1344  ax-11 1346  ax-4 1349  ax-17 1367  ax-i9 1371  ax-ial 1376  ax-i5r 1377  ax-ext 1928
This theorem depends on definitions:  df-bi 108  df-tru 1204  df-fal 1205  df-nf 1296  df-sb 1586  df-clab 1932  df-cleq 1938  df-clel 1941  df-nel 2110  df-v 2443
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