Theorem exnelv

Description: For any set x , there is a set not contained in x . The proof is based on Russell's paradox. (Contributed by NM, 23-Aug-1993) Remove use of ax-12 and ax-13 . (Revised by BJ, 31-May-2019) Extract from nalset . (Revised by Matthew House, 12-Apr-2026)

	Ref	Expression
Assertion	exnelv	⊢ ∃ 𝑦 ¬ 𝑦 ∈ 𝑥

Proof

Step	Hyp	Ref	Expression
1		ax-sep	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) )
2		elequ1	⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥 ) )
3		elequ2	⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ 𝑧 ↔ 𝑧 ∈ 𝑦 ) )
4	3	notbid	⊢ ( 𝑧 = 𝑦 → ( ¬ 𝑧 ∈ 𝑧 ↔ ¬ 𝑧 ∈ 𝑦 ) )
5	2 4	anbi12d	⊢ ( 𝑧 = 𝑦 → ( ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ↔ ( 𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦 ) ) )
6	5	bibi2d	⊢ ( 𝑧 = 𝑦 → ( ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ) ↔ ( 𝑧 ∈ 𝑦 ↔ ( 𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦 ) ) ) )
7		pclem6	⊢ ( ( 𝑧 ∈ 𝑦 ↔ ( 𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ¬ 𝑦 ∈ 𝑥 )
8	6 7	biimtrdi	⊢ ( 𝑧 = 𝑦 → ( ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ) → ¬ 𝑦 ∈ 𝑥 ) )
9	8	spimvw	⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ) → ¬ 𝑦 ∈ 𝑥 )
10	1 9	eximii	⊢ ∃ 𝑦 ¬ 𝑦 ∈ 𝑥