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	⊢ ¬ ∃ 𝑦 𝑦 = { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 }

Proof

Step	Hyp	Ref	Expression
1		ru0	⊢ ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ¬ 𝑧 ∈ 𝑧 )
2		id	⊢ ( 𝑥 = 𝑧 → 𝑥 = 𝑧 )
3	2 2	eleq12d	⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ 𝑥 ↔ 𝑧 ∈ 𝑧 ) )
4	3	notbid	⊢ ( 𝑥 = 𝑧 → ( ¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑧 ∈ 𝑧 ) )
5	4	eqabbw	⊢ ( 𝑦 = { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 } ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ¬ 𝑧 ∈ 𝑧 ) )
6	1 5	mtbir	⊢ ¬ 𝑦 = { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 }
7	6	nex	⊢ ¬ ∃ 𝑦 𝑦 = { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 }