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Theorem ru 2786
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 3903. (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 pm5.19 632 . . . . . 6 ¬ (𝑦𝑦 ↔ ¬ 𝑦𝑦)
2 eleq1 2116 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
3 df-nel 2315 . . . . . . . . 9 (𝑥𝑥 ↔ ¬ 𝑥𝑥)
4 id 19 . . . . . . . . . . 11 (𝑥 = 𝑦𝑥 = 𝑦)
54, 4eleq12d 2124 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
65notbid 602 . . . . . . . . 9 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
73, 6syl5bb 185 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑥 ↔ ¬ 𝑦𝑦))
82, 7bibi12d 228 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝑦𝑥𝑥) ↔ (𝑦𝑦 ↔ ¬ 𝑦𝑦)))
98spv 1756 . . . . . 6 (∀𝑥(𝑥𝑦𝑥𝑥) → (𝑦𝑦 ↔ ¬ 𝑦𝑦))
101, 9mto 598 . . . . 5 ¬ ∀𝑥(𝑥𝑦𝑥𝑥)
11 abeq2 2162 . . . . 5 (𝑦 = {𝑥𝑥𝑥} ↔ ∀𝑥(𝑥𝑦𝑥𝑥))
1210, 11mtbir 606 . . . 4 ¬ 𝑦 = {𝑥𝑥𝑥}
1312nex 1405 . . 3 ¬ ∃𝑦 𝑦 = {𝑥𝑥𝑥}
14 isset 2578 . . 3 ({𝑥𝑥𝑥} ∈ V ↔ ∃𝑦 𝑦 = {𝑥𝑥𝑥})
1513, 14mtbir 606 . 2 ¬ {𝑥𝑥𝑥} ∈ V
16 df-nel 2315 . 2 ({𝑥𝑥𝑥} ∉ V ↔ ¬ {𝑥𝑥𝑥} ∈ V)
1715, 16mpbir 138 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wnel 2314  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nel 2315  df-v 2576
This theorem is referenced by: (None)
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