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Theorem ssclaxsep 44975
Description: A class that is closed under subsets models the Axiom of Separation ax-sep 5294. 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 (∀𝑧𝑀 𝒫 𝑧𝑀 → ∀𝑧𝑀𝑦𝑀𝑥𝑀 (𝑥𝑦 ↔ (𝑥𝑧𝜑)))
Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧   𝑦,𝑀
Allowed substitution hints:   𝜑(𝑥)   𝑀(𝑥,𝑧)

Proof of Theorem ssclaxsep
StepHypRef Expression
1 ax-sep 5294 . . . 4 𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
2 biimp 215 . . . . . . . . . 10 ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) → (𝑥𝑦 → (𝑥𝑧𝜑)))
3 simpl 482 . . . . . . . . . 10 ((𝑥𝑧𝜑) → 𝑥𝑧)
42, 3syl6 35 . . . . . . . . 9 ((𝑥𝑦 ↔ (𝑥𝑧𝜑)) → (𝑥𝑦𝑥𝑧))
54alimi 1811 . . . . . . . 8 (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) → ∀𝑥(𝑥𝑦𝑥𝑧))
6 velpw 4603 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝑧𝑦𝑧)
7 df-ss 3967 . . . . . . . . 9 (𝑦𝑧 ↔ ∀𝑥(𝑥𝑦𝑥𝑧))
86, 7bitr2i 276 . . . . . . . 8 (∀𝑥(𝑥𝑦𝑥𝑧) ↔ 𝑦 ∈ 𝒫 𝑧)
95, 8sylib 218 . . . . . . 7 (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) → 𝑦 ∈ 𝒫 𝑧)
10 ssel 3976 . . . . . . 7 (𝒫 𝑧𝑀 → (𝑦 ∈ 𝒫 𝑧𝑦𝑀))
119, 10syl5 34 . . . . . 6 (𝒫 𝑧𝑀 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) → 𝑦𝑀))
12 alral 3074 . . . . . 6 (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) → ∀𝑥𝑀 (𝑥𝑦 ↔ (𝑥𝑧𝜑)))
1311, 12jca2 513 . . . . 5 (𝒫 𝑧𝑀 → (∀𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) → (𝑦𝑀 ∧ ∀𝑥𝑀 (𝑥𝑦 ↔ (𝑥𝑧𝜑)))))
1413eximdv 1917 . . . 4 (𝒫 𝑧𝑀 → (∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑)) → ∃𝑦(𝑦𝑀 ∧ ∀𝑥𝑀 (𝑥𝑦 ↔ (𝑥𝑧𝜑)))))
151, 14mpi 20 . . 3 (𝒫 𝑧𝑀 → ∃𝑦(𝑦𝑀 ∧ ∀𝑥𝑀 (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
16 df-rex 3070 . . 3 (∃𝑦𝑀𝑥𝑀 (𝑥𝑦 ↔ (𝑥𝑧𝜑)) ↔ ∃𝑦(𝑦𝑀 ∧ ∀𝑥𝑀 (𝑥𝑦 ↔ (𝑥𝑧𝜑))))
1715, 16sylibr 234 . 2 (𝒫 𝑧𝑀 → ∃𝑦𝑀𝑥𝑀 (𝑥𝑦 ↔ (𝑥𝑧𝜑)))
1817ralimi 3082 1 (∀𝑧𝑀 𝒫 𝑧𝑀 → ∀𝑧𝑀𝑦𝑀𝑥𝑀 (𝑥𝑦 ↔ (𝑥𝑧𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wex 1779  wcel 2108  wral 3060  wrex 3069  wss 3950  𝒫 cpw 4598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5294
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-v 3481  df-ss 3967  df-pw 4600
This theorem is referenced by:  wfaxsep  44988
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