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Theorem ru 2908
 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 , asserted that any collection of sets is a set i.e. belongs to the universe of all sets. In particular, by substituting (the "Russell class") for , it asserted , meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove . 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 is a set only when it is smaller than some other set . 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 4049. (Contributed by NM, 7-Aug-1994.)
Assertion
Ref Expression
ru

Proof of Theorem ru
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pm5.19 695 . . . . . 6
2 eleq1 2202 . . . . . . . 8
3 df-nel 2404 . . . . . . . . 9
4 id 19 . . . . . . . . . . 11
54, 4eleq12d 2210 . . . . . . . . . 10
65notbid 656 . . . . . . . . 9
73, 6syl5bb 191 . . . . . . . 8
82, 7bibi12d 234 . . . . . . 7
98spv 1832 . . . . . 6
101, 9mto 651 . . . . 5
11 abeq2 2248 . . . . 5
1210, 11mtbir 660 . . . 4
1312nex 1476 . . 3
14 isset 2692 . . 3
1513, 14mtbir 660 . 2
16 df-nel 2404 . 2
1715, 16mpbir 145 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 104  wal 1329   wceq 1331  wex 1468   wcel 1480  cab 2125   wnel 2403  cvv 2686 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 603  ax-in2 604  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nel 2404  df-v 2688 This theorem is referenced by: (None)
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