Theorem ruALT 4294

Description: Alternate proof of Russell's Paradox ru 2814, simplified using (indirectly) the Axiom of Set Induction ax-setind 4280. (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 3909	. . 3 |- -. _V e. _V
2		df-nel 2340	. . 3 |- ( _V e/ _V <-> -. _V e. _V )
3	1, 2	mpbir 144	. 2 |- _V e/ _V
4		ruv 4293	. . 3 |- { x | x e/ x } = _V
5		neleq1 2343	. . 3 |- ( { x | x e/ x } = _V -> ( { x | x e/ x } e/ _V <-> _V e/ _V ) )
6	4, 5	ax-mp 7	. 2 |- ( { x | x e/ x } e/ _V <-> _V e/ _V )
7	3, 6	mpbir 144	1 |- { x | x e/ x } e/ _V

Syntax hints:   -. wn 3   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067   e/ wnel 2339   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-v 2603  df-dif 2975  df-sn 3404

This theorem is referenced by: (None)