Theorem ru 3041

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 4228. (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		pm5.19 714	. . . . . 6 |- -. ( y e. y <-> -. y e. y )
2		eleq1 2295	. . . . . . . 8 |- ( x = y -> ( x e. y <-> y e. y ) )
3		df-nel 2508	. . . . . . . . 9 |- ( x e/ x <-> -. x e. x )
4		id 19	. . . . . . . . . . 11 |- ( x = y -> x = y )
5	4, 4	eleq12d 2303	. . . . . . . . . 10 |- ( x = y -> ( x e. x <-> y e. y ) )
6	5	notbid 673	. . . . . . . . 9 |- ( x = y -> ( -. x e. x <-> -. y e. y ) )
7	3, 6	bitrid 192	. . . . . . . 8 |- ( x = y -> ( x e/ x <-> -. y e. y ) )
8	2, 7	bibi12d 235	. . . . . . 7 |- ( x = y -> ( ( x e. y <-> x e/ x ) <-> ( y e. y <-> -. y e. y ) ) )
9	8	spv 1909	. . . . . 6 |- ( A. x ( x e. y <-> x e/ x ) -> ( y e. y <-> -. y e. y ) )
10	1, 9	mto 668	. . . . 5 |- -. A. x ( x e. y <-> x e/ x )
11		abeq2 2341	. . . . 5 |- ( y = { x | x e/ x } <-> A. x ( x e. y <-> x e/ x ) )
12	10, 11	mtbir 678	. . . 4 |- -. y = { x | x e/ x }
13	12	nex 1549	. . 3 |- -. E. y y = { x | x e/ x }
14		isset 2820	. . 3 |- ( { x | x e/ x } e. _V <-> E. y y = { x | x e/ x } )
15	13, 14	mtbir 678	. 2 |- -. { x | x e/ x } e. _V
16		df-nel 2508	. 2 |- ( { x | x e/ x } e/ _V <-> -. { x | x e/ x } e. _V )
17	15, 16	mpbir 146	1 |- { x | x e/ x } e/ _V

Syntax hints:   -. wn 3   <-> wb 105   A.wal 1396   = wceq 1398   E.wex 1541   e. wcel 2203   {cab 2218   e/ wnel 2507   _Vcvv 2813

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 619  ax-in2 620  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nel 2508  df-v 2815

This theorem is referenced by: (None)