Theorem ru 2853

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 3978. (Contributed by NM, 7-Aug-1994.)

Assertion

Ref	Expression
ru	⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Proof of Theorem ru

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm5.19 660	. . . . . 6 ⊢ ¬ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)
2		eleq1 2157	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦))
3		df-nel 2358	. . . . . . . . 9 ⊢ (𝑥 ∉ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥)
4		id 19	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
5	4, 4	eleq12d 2165	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
6	5	notbid 630	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
7	3, 6	syl5bb 191	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
8	2, 7	bibi12d 234	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) ↔ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)))
9	8	spv 1795	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥) → (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦))
10	1, 9	mto 626	. . . . 5 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥)
11		abeq2 2203	. . . . 5 ⊢ (𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥))
12	10, 11	mtbir 634	. . . 4 ⊢ ¬ 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥}
13	12	nex 1441	. . 3 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥}
14		isset 2639	. . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V ↔ ∃𝑦 𝑦 = {𝑥 ∣ 𝑥 ∉ 𝑥})
15	13, 14	mtbir 634	. 2 ⊢ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V
16		df-nel 2358	. 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ ¬ {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V)
17	15, 16	mpbir 145	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 104  ∀wal 1294   = wceq 1296  ∃wex 1433   ∈ wcel 1445  {cab 2081   ∉ wnel 2357  Vcvv 2633

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nel 2358  df-v 2635

This theorem is referenced by: (None)