Theorem ru 2861

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 e. _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 e/ x } (the "Russell class") for A, it asserted { x | x e/ x } e. _V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove { x | x e/ x } e/ _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 3986. (Contributed by NM, 7-Aug-1994.)

Assertion

Ref	Expression
ru	|- { x | x e/ x } e/ _V

Proof of Theorem ru

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm5.19 663	. . . . . 6 |- -. ( y e. y <-> -. y e. y )
2		eleq1 2162	. . . . . . . 8 |- ( x = y -> ( x e. y <-> y e. y ) )
3		df-nel 2363	. . . . . . . . 9 |- ( x e/ x <-> -. x e. x )
4		id 19	. . . . . . . . . . 11 |- ( x = y -> x = y )
5	4, 4	eleq12d 2170	. . . . . . . . . 10 |- ( x = y -> ( x e. x <-> y e. y ) )
6	5	notbid 633	. . . . . . . . 9 |- ( x = y -> ( -. x e. x <-> -. y e. y ) )
7	3, 6	syl5bb 191	. . . . . . . 8 |- ( x = y -> ( x e/ x <-> -. y e. y ) )
8	2, 7	bibi12d 234	. . . . . . 7 |- ( x = y -> ( ( x e. y <-> x e/ x ) <-> ( y e. y <-> -. y e. y ) ) )
9	8	spv 1799	. . . . . 6 |- ( A. x ( x e. y <-> x e/ x ) -> ( y e. y <-> -. y e. y ) )
10	1, 9	mto 629	. . . . 5 |- -. A. x ( x e. y <-> x e/ x )
11		abeq2 2208	. . . . 5 |- ( y = { x | x e/ x } <-> A. x ( x e. y <-> x e/ x ) )
12	10, 11	mtbir 637	. . . 4 |- -. y = { x | x e/ x }
13	12	nex 1444	. . 3 |- -. E. y y = { x | x e/ x }
14		isset 2647	. . 3 |- ( { x | x e/ x } e. _V <-> E. y y = { x | x e/ x } )
15	13, 14	mtbir 637	. 2 |- -. { x | x e/ x } e. _V
16		df-nel 2363	. 2 |- ( { x | x e/ x } e/ _V <-> -. { x | x e/ x } e. _V )
17	15, 16	mpbir 145	1 |- { x | x e/ x } e/ _V

Syntax hints:   -. wn 3   <-> wb 104   A.wal 1297   = wceq 1299   E.wex 1436   e. wcel 1448   {cab 2086   e/ wnel 2362   _Vcvv 2641

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 584  ax-in2 585  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nel 2363  df-v 2643

This theorem is referenced by: (None)