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Theorem ru 2976
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 4136. (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 pm5.19 707 . . . . . 6 ¬ (𝑦𝑦 ↔ ¬ 𝑦𝑦)
2 eleq1 2252 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
3 df-nel 2456 . . . . . . . . 9 (𝑥𝑥 ↔ ¬ 𝑥𝑥)
4 id 19 . . . . . . . . . . 11 (𝑥 = 𝑦𝑥 = 𝑦)
54, 4eleq12d 2260 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
65notbid 668 . . . . . . . . 9 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
73, 6bitrid 192 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑥 ↔ ¬ 𝑦𝑦))
82, 7bibi12d 235 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝑦𝑥𝑥) ↔ (𝑦𝑦 ↔ ¬ 𝑦𝑦)))
98spv 1871 . . . . . 6 (∀𝑥(𝑥𝑦𝑥𝑥) → (𝑦𝑦 ↔ ¬ 𝑦𝑦))
101, 9mto 663 . . . . 5 ¬ ∀𝑥(𝑥𝑦𝑥𝑥)
11 abeq2 2298 . . . . 5 (𝑦 = {𝑥𝑥𝑥} ↔ ∀𝑥(𝑥𝑦𝑥𝑥))
1210, 11mtbir 672 . . . 4 ¬ 𝑦 = {𝑥𝑥𝑥}
1312nex 1511 . . 3 ¬ ∃𝑦 𝑦 = {𝑥𝑥𝑥}
14 isset 2758 . . 3 ({𝑥𝑥𝑥} ∈ V ↔ ∃𝑦 𝑦 = {𝑥𝑥𝑥})
1513, 14mtbir 672 . 2 ¬ {𝑥𝑥𝑥} ∈ V
16 df-nel 2456 . 2 ({𝑥𝑥𝑥} ∉ V ↔ ¬ {𝑥𝑥𝑥} ∈ V)
1715, 16mpbir 146 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 105  wal 1362   = wceq 1364  wex 1503  wcel 2160  {cab 2175  wnel 2455  Vcvv 2752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nel 2456  df-v 2754
This theorem is referenced by: (None)
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