Theorem pwclaxpow 44947

Description: Suppose 𝑀 is a transitive class that is closed under power sets intersected with 𝑀. Then, 𝑀 models the Axiom of Power Sets ax-pow 5315. One direction of Lemma II.2.8 of [Kunen2] p. 113. (Contributed by Eric Schmidt, 19-Oct-2025.)

Assertion

Ref	Expression
pwclaxpow	⊢ ((Tr 𝑀 ∧ ∀𝑥 ∈ 𝑀 (𝒫 𝑥 ∩ 𝑀) ∈ 𝑀) → ∀𝑥 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑀

Proof of Theorem pwclaxpow

Step	Hyp	Ref	Expression
1		velpw 4564	. . . . . . . 8 ⊢ (𝑧 ∈ 𝒫 𝑥 ↔ 𝑧 ⊆ 𝑥)
2		ssabso 44937	. . . . . . . 8 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → (𝑧 ⊆ 𝑥 ↔ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)))
3	1, 2	bitrid 283	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → (𝑧 ∈ 𝒫 𝑥 ↔ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)))
4		elin 3927	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝒫 𝑥 ∩ 𝑀) ↔ (𝑧 ∈ 𝒫 𝑥 ∧ 𝑧 ∈ 𝑀))
5	4	simplbi2com 502	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑀 → (𝑧 ∈ 𝒫 𝑥 → 𝑧 ∈ (𝒫 𝑥 ∩ 𝑀)))
6	5	adantl 481	. . . . . . 7 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → (𝑧 ∈ 𝒫 𝑥 → 𝑧 ∈ (𝒫 𝑥 ∩ 𝑀)))
7	3, 6	sylbird 260	. . . . . 6 ⊢ ((Tr 𝑀 ∧ 𝑧 ∈ 𝑀) → (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ (𝒫 𝑥 ∩ 𝑀)))
8	7	ralrimiva 3125	. . . . 5 ⊢ (Tr 𝑀 → ∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ (𝒫 𝑥 ∩ 𝑀)))
9		eleq2 2817	. . . . . . . 8 ⊢ (𝑦 = (𝒫 𝑥 ∩ 𝑀) → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ (𝒫 𝑥 ∩ 𝑀)))
10	9	imbi2d 340	. . . . . . 7 ⊢ (𝑦 = (𝒫 𝑥 ∩ 𝑀) → ((∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ (𝒫 𝑥 ∩ 𝑀))))
11	10	ralbidv 3156	. . . . . 6 ⊢ (𝑦 = (𝒫 𝑥 ∩ 𝑀) → (∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ (𝒫 𝑥 ∩ 𝑀))))
12	11	rspcev 3585	. . . . 5 ⊢ (((𝒫 𝑥 ∩ 𝑀) ∈ 𝑀 ∧ ∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ (𝒫 𝑥 ∩ 𝑀))) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
13	8, 12	sylan2 593	. . . 4 ⊢ (((𝒫 𝑥 ∩ 𝑀) ∈ 𝑀 ∧ Tr 𝑀) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
14	13	expcom 413	. . 3 ⊢ (Tr 𝑀 → ((𝒫 𝑥 ∩ 𝑀) ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)))
15	14	ralimdv 3147	. 2 ⊢ (Tr 𝑀 → (∀𝑥 ∈ 𝑀 (𝒫 𝑥 ∩ 𝑀) ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)))
16	15	imp 406	1 ⊢ ((Tr 𝑀 ∧ ∀𝑥 ∈ 𝑀 (𝒫 𝑥 ∩ 𝑀) ∈ 𝑀) → ∀𝑥 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  Tr wtr 5209

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-v 3446  df-in 3918  df-ss 3928  df-pw 4561  df-uni 4868  df-tr 5210

This theorem is referenced by:  wfaxpow  44960