Theorem ruALT 4528

Description: Alternate proof of Russell's Paradox ru 2950, simplified using (indirectly) the Axiom of Set Induction ax-setind 4514. (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 4114	. . 3 |- -. _V e. _V
2		df-nel 2432	. . 3 |- ( _V e/ _V <-> -. _V e. _V )
3	1, 2	mpbir 145	. 2 |- _V e/ _V
4		ruv 4527	. . 3 |- { x | x e/ x } = _V
5		neleq1 2435	. . 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 1343   e. wcel 2136   {cab 2151   e/ wnel 2431   _Vcvv 2726

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-v 2728  df-dif 3118  df-sn 3582

This theorem is referenced by: (None)