Theorem zfregcl 9499

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 2152 and ax-12 2189. (Revised by TM, 31-Dec-2025.)

Assertion

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

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem zfregcl

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2828	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝐴))
2	1	exbidv 1928	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 𝑥 ∈ 𝑧 ↔ ∃𝑥 𝑥 ∈ 𝐴))
3		eleq2 2828	. . . . . 6 ⊢ (𝑧 = 𝐴 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝐴))
4	3	notbid 319	. . . . 5 ⊢ (𝑧 = 𝐴 → (¬ 𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝐴))
5	4	ralbidv 3162	. . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))
6	5	rexeqbi1dv 3308	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))
7	2, 6	imbi12d 345	. 2 ⊢ (𝑧 = 𝐴 → ((∃𝑥 𝑥 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧) ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴)))
8		ax-reg 9497	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝑧 → ∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧)))
9		df-ral 3054	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧))
10	9	rexbii 3086	. . . 4 ⊢ (∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑧 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧))
11		df-rex 3064	. . . 4 ⊢ (∃𝑥 ∈ 𝑧 ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧) ↔ ∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧)))
12	10, 11	bitr2i 277	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝑧 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑧)) ↔ ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧)
13	8, 12	sylib 219	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑧)
14	7, 13	vtoclg 3500	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-reg 9497

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064

This theorem is referenced by:  zfreg  9501  elirrvOLDOLD  9504