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Theorem ru 2760
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 3869. (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 pm5.19 622 . . . . . 6 ¬ (y y ↔ ¬ y y)
2 eleq1 2100 . . . . . . . 8 (x = y → (x yy y))
3 df-nel 2207 . . . . . . . . 9 (xx ↔ ¬ x x)
4 id 19 . . . . . . . . . . 11 (x = yx = y)
54, 4eleq12d 2108 . . . . . . . . . 10 (x = y → (x xy y))
65notbid 592 . . . . . . . . 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 1740 . . . . . 6 (x(x yxx) → (y y ↔ ¬ y y))
101, 9mto 588 . . . . 5 ¬ x(x yxx)
11 abeq2 2146 . . . . 5 (y = {xxx} ↔ x(x yxx))
1210, 11mtbir 596 . . . 4 ¬ y = {xxx}
1312nex 1389 . . 3 ¬ y y = {xxx}
14 isset 2558 . . 3 ({xxx} V ↔ y y = {xxx})
1513, 14mtbir 596 . 2 ¬ {xxx} V
16 df-nel 2207 . 2 ({xxx} ∉ V ↔ ¬ {xxx} V)
1715, 16mpbir 134 1 {xxx} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98  wal 1241   = wceq 1243  wex 1381   wcel 1393  {cab 2026  wnel 2205  Vcvv 2554
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 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nel 2207  df-v 2556
This theorem is referenced by: (None)
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