Theorem pwelg 44670

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 490	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → 𝒫 𝑥 ∈ 𝐵)
2	1	ralimi 3099	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 𝒫 𝑥 ∈ 𝐵)
3		pweq 4570	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
4	3	eleq1d 2845	. . . 4 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵))
5	4	rspccv 3573	. . 3 ⊢ (∀𝑥 ∈ 𝐵 𝒫 𝑥 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝐵))
6	2, 5	syl 18	. 2 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝐵))
7		simpl 488	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐵)
8	7	ralimi 3099	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∪ 𝑥 ∈ 𝐵)
9		unieq 4877	. . . . . 6 ⊢ (𝑥 = 𝒫 𝐴 → ∪ 𝑥 = ∪ 𝒫 𝐴)
10		unipw 5418	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
11	9, 10	eqtrdi 2811	. . . . 5 ⊢ (𝑥 = 𝒫 𝐴 → ∪ 𝑥 = 𝐴)
12	11	eleq1d 2845	. . . 4 ⊢ (𝑥 = 𝒫 𝐴 → (∪ 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
13	12	rspccv 3573	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∪ 𝑥 ∈ 𝐵 → (𝒫 𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐵))
14	8, 13	syl 18	. 2 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝒫 𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐵))
15	6, 14	impbid 215	1 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  𝒫 cpw 4556  ∪ cuni 4866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3897  df-un 4150  df-pw 4558  df-sn 4585  df-pr 4587  df-uni 4867

This theorem is used by:  pwinfig  44671