Theorem ruALT 4474

Description: Alternate proof of Russell's Paradox ru 2912, simplified using (indirectly) the Axiom of Set Induction ax-setind 4460. (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 4068	. . 3 |- -. _V e. _V
2		df-nel 2405	. . 3 |- ( _V e/ _V <-> -. _V e. _V )
3	1, 2	mpbir 145	. 2 |- _V e/ _V
4		ruv 4473	. . 3 |- { x | x e/ x } = _V
5		neleq1 2408	. . 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 1332   e. wcel 1481   {cab 2126   e/ wnel 2404   _Vcvv 2689

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-v 2691  df-dif 3078  df-sn 3538

This theorem is referenced by: (None)