Theorem ruALT 4693

Description: Alternate proof of Russell's Paradox ru 3050, simplified using (indirectly) the Axiom of Set Induction ax-setind 4679. (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 4260	. . 3 |- -. _V e. _V
2		df-nel 2516	. . 3 |- ( _V e/ _V <-> -. _V e. _V )
3	1, 2	mpbir 146	. 2 |- _V e/ _V
4		ruv 4692	. . 3 |- { x | x e/ x } = _V
5		neleq1 2519	. . 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 1402   e. wcel 2209   {cab 2224   e/ wnel 2515   _Vcvv 2821

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-v 2823  df-dif 3222  df-sn 3711

This theorem is referenced by: (None)