Theorem pwelg 39912

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 487	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → 𝒫 𝑥 ∈ 𝐵)
2	1	ralimi 3160	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 𝒫 𝑥 ∈ 𝐵)
3		pweq 4541	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
4	3	eleq1d 2897	. . . 4 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵))
5	4	rspccv 3619	. . 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 485	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐵)
8	7	ralimi 3160	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∪ 𝑥 ∈ 𝐵)
9		unieq 4839	. . . . . 6 ⊢ (𝑥 = 𝒫 𝐴 → ∪ 𝑥 = ∪ 𝒫 𝐴)
10		unipw 5334	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
11	9, 10	syl6eq 2872	. . . . 5 ⊢ (𝑥 = 𝒫 𝐴 → ∪ 𝑥 = 𝐴)
12	11	eleq1d 2897	. . . 4 ⊢ (𝑥 = 𝒫 𝐴 → (∪ 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
13	12	rspccv 3619	. . 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 214	1 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  𝒫 cpw 4538  ∪ cuni 4831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4561  df-pr 4563  df-uni 4832

This theorem is referenced by:  pwinfig  39913