Theorem ru 2757

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 3866. (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		pm5.19 621	. . . . . 6 ⊢ ¬ (y ∈ y ↔ ¬ y ∈ y)
2		eleq1 2097	. . . . . . . 8 ⊢ (x = y → (x ∈ y ↔ y ∈ y))
3		df-nel 2204	. . . . . . . . 9 ⊢ (x ∉ x ↔ ¬ x ∈ x)
4		id 19	. . . . . . . . . . 11 ⊢ (x = y → x = y)
5	4, 4	eleq12d 2105	. . . . . . . . . 10 ⊢ (x = y → (x ∈ x ↔ y ∈ y))
6	5	notbid 591	. . . . . . . . 9 ⊢ (x = y → (¬ x ∈ x ↔ ¬ y ∈ y))
7	3, 6	syl5bb 181	. . . . . . . 8 ⊢ (x = y → (x ∉ x ↔ ¬ y ∈ y))
8	2, 7	bibi12d 224	. . . . . . 7 ⊢ (x = y → ((x ∈ y ↔ x ∉ x) ↔ (y ∈ y ↔ ¬ y ∈ y)))
9	8	spv 1737	. . . . . 6 ⊢ (∀x(x ∈ y ↔ x ∉ x) → (y ∈ y ↔ ¬ y ∈ y))
10	1, 9	mto 587	. . . . 5 ⊢ ¬ ∀x(x ∈ y ↔ x ∉ x)
11		abeq2 2143	. . . . 5 ⊢ (y = {x ∣ x ∉ x} ↔ ∀x(x ∈ y ↔ x ∉ x))
12	10, 11	mtbir 595	. . . 4 ⊢ ¬ y = {x ∣ x ∉ x}
13	12	nex 1386	. . 3 ⊢ ¬ ∃y y = {x ∣ x ∉ x}
14		isset 2555	. . 3 ⊢ ({x ∣ x ∉ x} ∈ V ↔ ∃y y = {x ∣ x ∉ x})
15	13, 14	mtbir 595	. 2 ⊢ ¬ {x ∣ x ∉ x} ∈ V
16		df-nel 2204	. 2 ⊢ ({x ∣ x ∉ x} ∉ V ↔ ¬ {x ∣ x ∉ x} ∈ V)
17	15, 16	mpbir 134	1 ⊢ {x ∣ x ∉ x} ∉ V

Syntax hints:  ¬ wn 3   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023   ∉ wnel 2202  Vcvv 2551

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nel 2204  df-v 2553

This theorem is referenced by: (None)