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Theorem pwelg 36884
 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 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem pwelg
StepHypRef Expression
1 simpr 476 . . . 4 (( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → 𝒫 𝑥𝐵)
21ralimi 2936 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ∀𝑥𝐵 𝒫 𝑥𝐵)
3 pweq 4111 . . . . 5 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
43eleq1d 2672 . . . 4 (𝑥 = 𝐴 → (𝒫 𝑥𝐵 ↔ 𝒫 𝐴𝐵))
54rspccv 3279 . . 3 (∀𝑥𝐵 𝒫 𝑥𝐵 → (𝐴𝐵 → 𝒫 𝐴𝐵))
62, 5syl 17 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 → 𝒫 𝐴𝐵))
7 simpl 472 . . . 4 (( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → 𝑥𝐵)
87ralimi 2936 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ∀𝑥𝐵 𝑥𝐵)
9 unieq 4380 . . . . . 6 (𝑥 = 𝒫 𝐴 𝑥 = 𝒫 𝐴)
10 unipw 4845 . . . . . 6 𝒫 𝐴 = 𝐴
119, 10syl6eq 2660 . . . . 5 (𝑥 = 𝒫 𝐴 𝑥 = 𝐴)
1211eleq1d 2672 . . . 4 (𝑥 = 𝒫 𝐴 → ( 𝑥𝐵𝐴𝐵))
1312rspccv 3279 . . 3 (∀𝑥𝐵 𝑥𝐵 → (𝒫 𝐴𝐵𝐴𝐵))
148, 13syl 17 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝒫 𝐴𝐵𝐴𝐵))
156, 14impbid 201 1 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  𝒫 cpw 4108  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373 This theorem is referenced by:  pwinfig  36885
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