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	$$⊢\neg \exists yy=\left\{x|\neg x\in x\right\}$$

Proof

Step	Hyp	Ref	Expression
1		ru0	$$⊢\neg \forall z\left(z\in y\leftrightarrow \neg z\in z\right)$$
2		id	$$⊢x=z\to x=z$$
3	2 2	eleq12d	$$⊢x=z\to \left(x\in x\leftrightarrow z\in z\right)$$
4	3	notbid	$$⊢x=z\to \left(\neg x\in x\leftrightarrow \neg z\in z\right)$$
5	4	eqabbw	$$⊢y=\left\{x|\neg x\in x\right\}\leftrightarrow \forall z\left(z\in y\leftrightarrow \neg z\in z\right)$$
6	1 5	mtbir	$$⊢\neg y=\left\{x|\neg x\in x\right\}$$
7	6	nex	$$⊢\neg \exists yy=\left\{x|\neg x\in x\right\}$$