Theorem ruALT 4673

Description: Alternate proof of Russell's Paradox ru 3041, simplified using (indirectly) the Axiom of Set Induction ax-setind 4659. (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 4242	. . 3 |- -. _V e. _V
2		df-nel 2508	. . 3 |- ( _V e/ _V <-> -. _V e. _V )
3	1, 2	mpbir 146	. 2 |- _V e/ _V
4		ruv 4672	. . 3 |- { x | x e/ x } = _V
5		neleq1 2511	. . 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 146	1 |- { x | x e/ x } e/ _V

Syntax hints:   -. wn 3   <-> wb 105   = wceq 1398   e. wcel 2203   {cab 2218   e/ wnel 2507   _Vcvv 2813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-v 2815  df-dif 3213  df-sn 3695

This theorem is referenced by: (None)