Theorem bj-ru1

Description: A version of Russell's paradox ru not mentioning the universal class. (see also bj-ru ). (Contributed by BJ, 12-Oct-2019) Remove usage of ax-10 , ax-11 , ax-12 by using eqabbw following BTernaryTau's similar revision of ru . (Revised by BJ, 28-Jun-2025) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ru1	|- -. E. y y = { x | -. x e. x }

Proof

Step	Hyp	Ref	Expression
1		ru0	|- -. A. z ( z e. y <-> -. z e. z )
2		id	|- ( x = z -> x = z )
3	2 2	eleq12d	|- ( x = z -> ( x e. x <-> z e. z ) )
4	3	notbid	|- ( x = z -> ( -. x e. x <-> -. z e. z ) )
5	4	eqabbw	|- ( y = { x | -. x e. x } <-> A. z ( z e. y <-> -. z e. z ) )
6	1 5	mtbir	|- -. y = { x | -. x e. x }
7	6	nex	|- -. E. y y = { x | -. x e. x }