Theorem ruALT 4461

Description: Alternate proof of Russell's Paradox ru 2903, simplified using (indirectly) the Axiom of Set Induction ax-setind 4447. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ruALT

Step	Hyp	Ref	Expression
1		vprc 4055	. . 3 |- -. _V e. _V
2		df-nel 2402	. . 3 |- ( _V e/ _V <-> -. _V e. _V )
3	1, 2	mpbir 145	. 2 |- _V e/ _V
4		ruv 4460	. . 3 |- { x | x e/ x } = _V
5		neleq1 2405	. . 3 |- ( { x | x e/ x } = _V -> ( { x | x e/ x } e/ _V <-> _V e/ _V ) )
6	4, 5	ax-mp 5	. 2 |- ( { x | x e/ x } e/ _V <-> _V e/ _V )
7	3, 6	mpbir 145	1 |- { x | x e/ x } e/ _V

Syntax hints:   -. wn 3   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2123   e/ wnel 2401   _Vcvv 2681

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-v 2683  df-dif 3068  df-sn 3528

This theorem is referenced by: (None)