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Theorem ru 2740
 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 {x ∣ x ∉ x} (the "Russell class") for A, it asserted {x ∣ x ∉ x} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {x ∣ x ∉ x} ∉ 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 3849. (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 2082 . . . . . . . 8 (x = y → (x yy y))
3 df-nel 2189 . . . . . . . . 9 (xx ↔ ¬ x x)
4 id 19 . . . . . . . . . . 11 (x = yx = y)
54, 4eleq12d 2090 . . . . . . . . . 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 1722 . . . . . 6 (x(x yxx) → (y y ↔ ¬ y y))
101, 9mto 575 . . . . 5 ¬ x(x yxx)
11 abeq2 2128 . . . . 5 (y = {xxx} ↔ x(x yxx))
1210, 11mtbir 583 . . . 4 ¬ y = {xxx}
1312nex 1370 . . 3 ¬ y y = {xxx}
14 isset 2539 . . 3 ({xxx} V ↔ y y = {xxx})
1513, 14mtbir 583 . 2 ¬ {xxx} V
16 df-nel 2189 . 2 ({xxx} ∉ V ↔ ¬ {xxx} V)
1715, 16mpbir 134 1 {xxx} ∉ V
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98  ∀wal 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  {cab 2008   ∉ wnel 2187  Vcvv 2535 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nel 2189  df-v 2537 This theorem is referenced by: (None)
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