Theorem rusbcALT 38640

Description: A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
rusbcALT	|- { x | x e/ x } e/ _V

Proof of Theorem rusbcALT

Step	Hyp	Ref	Expression
1		pm5.19 375	. . 3 |- -. ( { x | x e/ x } e. { x | x e/ x } <-> -. { x | x e/ x } e. { x | x e/ x } )
2		sbcnel12g 3985	. . . 4 |- ( { x | x e/ x } e. _V -> ( [. { x | x e/ x } / x ]. x e/ x <-> [_ { x | x e/ x } / x ]_ x e/ [_ { x | x e/ x } / x ]_ x ) )
3		sbc8g 3443	. . . 4 |- ( { x | x e/ x } e. _V -> ( [. { x | x e/ x } / x ]. x e/ x <-> { x | x e/ x } e. { x | x e/ x } ) )
4		df-nel 2898	. . . . 5 |- ( [_ { x | x e/ x } / x ]_ x e/ [_ { x | x e/ x } / x ]_ x <-> -. [_ { x | x e/ x } / x ]_ x e. [_ { x | x e/ x } / x ]_ x )
5		csbvarg 4003	. . . . . . 7 |- ( { x | x e/ x } e. _V -> [_ { x | x e/ x } / x ]_ x = { x | x e/ x } )
6	5, 5	eleq12d 2695	. . . . . 6 |- ( { x | x e/ x } e. _V -> ( [_ { x | x e/ x } / x ]_ x e. [_ { x | x e/ x } / x ]_ x <-> { x | x e/ x } e. { x | x e/ x } ) )
7	6	notbid 308	. . . . 5 |- ( { x | x e/ x } e. _V -> ( -. [_ { x | x e/ x } / x ]_ x e. [_ { x | x e/ x } / x ]_ x <-> -. { x | x e/ x } e. { x | x e/ x } ) )
8	4, 7	syl5bb 272	. . . 4 |- ( { x | x e/ x } e. _V -> ( [_ { x | x e/ x } / x ]_ x e/ [_ { x | x e/ x } / x ]_ x <-> -. { x | x e/ x } e. { x | x e/ x } ) )
9	2, 3, 8	3bitr3d 298	. . 3 |- ( { x | x e/ x } e. _V -> ( { x | x e/ x } e. { x | x e/ x } <-> -. { x | x e/ x } e. { x | x e/ x } ) )
10	1, 9	mto 188	. 2 |- -. { x | x e/ x } e. _V
11		df-nel 2898	. 2 |- ( { x | x e/ x } e/ _V <-> -. { x | x e/ x } e. _V )
12	10, 11	mpbir 221	1 |- { x | x e/ x } e/ _V

Syntax hints:   -. wn 3   <-> wb 196   e. wcel 1990   {cab 2608   e/ wnel 2897   _Vcvv 3200   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)