Theorem zfregcl

Description: The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021) Avoid ax-10 and ax-12 . (Revised by TM, 31-Dec-2025)

	Ref	Expression
Assertion	zfregcl	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 𝑥 ∈ 𝐴 → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝑧 = 𝐴 → ( 𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐴 ) )
2	1	exbidv	⊢ ( 𝑧 = 𝐴 → ( ∃ 𝑥 𝑥 ∈ 𝑧 ↔ ∃ 𝑥 𝑥 ∈ 𝐴 ) )
3		eleq2	⊢ ( 𝑧 = 𝐴 → ( 𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝐴 ) )
4	3	notbid	⊢ ( 𝑧 = 𝐴 → ( ¬ 𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝐴 ) )
5	4	ralbidv	⊢ ( 𝑧 = 𝐴 → ( ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴 ) )
6	5	rexeqbi1dv	⊢ ( 𝑧 = 𝐴 → ( ∃ 𝑥 ∈ 𝑧 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴 ) )
7	2 6	imbi12d	⊢ ( 𝑧 = 𝐴 → ( ( ∃ 𝑥 𝑥 ∈ 𝑧 → ∃ 𝑥 ∈ 𝑧 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ) ↔ ( ∃ 𝑥 𝑥 ∈ 𝐴 → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴 ) ) )
8		ax-reg	⊢ ( ∃ 𝑥 𝑥 ∈ 𝑧 → ∃ 𝑥 ( 𝑥 ∈ 𝑧 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧 ) ) )
9		df-ral	⊢ ( ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧 ) )
10	9	rexbii	⊢ ( ∃ 𝑥 ∈ 𝑧 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∃ 𝑥 ∈ 𝑧 ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧 ) )
11		df-rex	⊢ ( ∃ 𝑥 ∈ 𝑧 ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑧 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧 ) ) )
12	10 11	bitr2i	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝑧 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧 ) ) ↔ ∃ 𝑥 ∈ 𝑧 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 )
13	8 12	sylib	⊢ ( ∃ 𝑥 𝑥 ∈ 𝑧 → ∃ 𝑥 ∈ 𝑧 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 )
14	7 13	vtoclg	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 𝑥 ∈ 𝐴 → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴 ) )