Theorem pwelg 43550

Description: The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.)

Assertion

Ref	Expression
pwelg	⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem pwelg

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → 𝒫 𝑥 ∈ 𝐵)
2	1	ralimi 3072	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 𝒫 𝑥 ∈ 𝐵)
3		pweq 4594	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
4	3	eleq1d 2818	. . . 4 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵))
5	4	rspccv 3602	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝒫 𝑥 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝐵))
6	2, 5	syl 17	. 2 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝐵))
7		simpl 482	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐵)
8	7	ralimi 3072	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∪ 𝑥 ∈ 𝐵)
9		unieq 4898	. . . . . 6 ⊢ (𝑥 = 𝒫 𝐴 → ∪ 𝑥 = ∪ 𝒫 𝐴)
10		unipw 5435	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
11	9, 10	eqtrdi 2785	. . . . 5 ⊢ (𝑥 = 𝒫 𝐴 → ∪ 𝑥 = 𝐴)
12	11	eleq1d 2818	. . . 4 ⊢ (𝑥 = 𝒫 𝐴 → (∪ 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
13	12	rspccv 3602	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∪ 𝑥 ∈ 𝐵 → (𝒫 𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐵))
14	8, 13	syl 17	. 2 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝒫 𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐵))
15	6, 14	impbid 212	1 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  𝒫 cpw 4580  ∪ cuni 4887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-v 3465  df-un 3936  df-ss 3948  df-pw 4582  df-sn 4607  df-pr 4609  df-uni 4888

This theorem is referenced by:  pwinfig  43551