Theorem ruALT 4587

Description: Alternate proof of Russell's Paradox ru 2988, simplified using (indirectly) the Axiom of Set Induction ax-setind 4573. (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 4165	. . 3 |- -. _V e. _V
2		df-nel 2463	. . 3 |- ( _V e/ _V <-> -. _V e. _V )
3	1, 2	mpbir 146	. 2 |- _V e/ _V
4		ruv 4586	. . 3 |- { x | x e/ x } = _V
5		neleq1 2466	. . 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 1364   e. wcel 2167   {cab 2182   e/ wnel 2462   _Vcvv 2763

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-v 2765  df-dif 3159  df-sn 3628

This theorem is referenced by: (None)