Theorem ru 2945

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 4094. (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		pm5.19 696	. . . . . 6 |- -. ( y e. y <-> -. y e. y )
2		eleq1 2227	. . . . . . . 8 |- ( x = y -> ( x e. y <-> y e. y ) )
3		df-nel 2430	. . . . . . . . 9 |- ( x e/ x <-> -. x e. x )
4		id 19	. . . . . . . . . . 11 |- ( x = y -> x = y )
5	4, 4	eleq12d 2235	. . . . . . . . . 10 |- ( x = y -> ( x e. x <-> y e. y ) )
6	5	notbid 657	. . . . . . . . 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 1847	. . . . . 6 |- ( A. x ( x e. y <-> x e/ x ) -> ( y e. y <-> -. y e. y ) )
10	1, 9	mto 652	. . . . 5 |- -. A. x ( x e. y <-> x e/ x )
11		abeq2 2273	. . . . 5 |- ( y = { x | x e/ x } <-> A. x ( x e. y <-> x e/ x ) )
12	10, 11	mtbir 661	. . . 4 |- -. y = { x | x e/ x }
13	12	nex 1487	. . 3 |- -. E. y y = { x | x e/ x }
14		isset 2727	. . 3 |- ( { x | x e/ x } e. _V <-> E. y y = { x | x e/ x } )
15	13, 14	mtbir 661	. 2 |- -. { x | x e/ x } e. _V
16		df-nel 2430	. 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 1340   = wceq 1342   E.wex 1479   e. wcel 2135   {cab 2150   e/ wnel 2429   _Vcvv 2721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nel 2430  df-v 2723

This theorem is referenced by: (None)