Theorem ssclaxsep 45661

Description: A class that is closed under subsets models the Axiom of Separation ax-sep 5256. Lemma II.2.4(3) of [Kunen2] p. 111.

Note that, to obtain the relativization of an instance of Separation to 𝑀, the formula 𝜑 would need to be replaced with its relativization to 𝑀. However, this new formula is a valid substitution for 𝜑, so this theorem does establish that all instances of Separation hold in 𝑀. (Contributed by Eric Schmidt, 29-Sep-2025.)

Assertion

Ref	Expression
ssclaxsep	⊢ (∀𝑧 ∈ 𝑀 𝒫 𝑧 ⊆ 𝑀 → ∀𝑧 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧   𝑦,𝑀

Allowed substitution hints:   𝜑(𝑥)   𝑀(𝑥,𝑧)

Proof of Theorem ssclaxsep

Step	Hyp	Ref	Expression
1		ax-sep 5256	. . . 4 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
2		biimp 218	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑧 ∧ 𝜑)))
3		simpl 487	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝜑) → 𝑥 ∈ 𝑧)
4	2, 3	syl6 36	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → (𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧))
5	4	alimi 1839	. . . . . . . 8 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧))
6		velpw 4566	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝑧 ↔ 𝑦 ⊆ 𝑧)
7		df-ss 3921	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝑧 ↔ ∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧))
8	6, 7	bitr2i 279	. . . . . . . 8 ⊢ (∀𝑥(𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) ↔ 𝑦 ∈ 𝒫 𝑧)
9	5, 8	sylib 221	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → 𝑦 ∈ 𝒫 𝑧)
10		ssel 3930	. . . . . . 7 ⊢ (𝒫 𝑧 ⊆ 𝑀 → (𝑦 ∈ 𝒫 𝑧 → 𝑦 ∈ 𝑀))
11	9, 10	syl5 35	. . . . . 6 ⊢ (𝒫 𝑧 ⊆ 𝑀 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → 𝑦 ∈ 𝑀))
12		alral 3092	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
13	11, 12	jca2 522	. . . . 5 ⊢ (𝒫 𝑧 ⊆ 𝑀 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → (𝑦 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))))
14	13	eximdv 1945	. . . 4 ⊢ (𝒫 𝑧 ⊆ 𝑀 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → ∃𝑦(𝑦 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))))
15	1, 14	mpi 21	. . 3 ⊢ (𝒫 𝑧 ⊆ 𝑀 → ∃𝑦(𝑦 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
16		df-rex 3088	. . 3 ⊢ (∃𝑦 ∈ 𝑀 ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ ∃𝑦(𝑦 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
17	15, 16	sylibr 237	. 2 ⊢ (𝒫 𝑧 ⊆ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
18	17	ralimi 3100	1 ⊢ (∀𝑧 ∈ 𝑀 𝒫 𝑧 ⊆ 𝑀 → ∀𝑧 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566  ∃wex 1807   ∈ wcel 2141  ∀wral 3077  ∃wrex 3087   ⊆ wss 3904  𝒫 cpw 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-v 3455  df-ss 3921  df-pw 4563

This theorem is referenced by:  wfaxsep  45674