Theorem ruALT 4617

Description: Alternate proof of Russell's Paradox ru 3004, simplified using (indirectly) the Axiom of Set Induction ax-setind 4603. (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 4192	. . 3 |- -. _V e. _V
2		df-nel 2474	. . 3 |- ( _V e/ _V <-> -. _V e. _V )
3	1, 2	mpbir 146	. 2 |- _V e/ _V
4		ruv 4616	. . 3 |- { x | x e/ x } = _V
5		neleq1 2477	. . 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 1373   e. wcel 2178   {cab 2193   e/ wnel 2473   _Vcvv 2776

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-v 2778  df-dif 3176  df-sn 3649

This theorem is referenced by: (None)