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Theorem pwelg 40175
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 488 . . . 4 (( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → 𝒫 𝑥𝐵)
21ralimi 3155 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ∀𝑥𝐵 𝒫 𝑥𝐵)
3 pweq 4538 . . . . 5 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
43eleq1d 2900 . . . 4 (𝑥 = 𝐴 → (𝒫 𝑥𝐵 ↔ 𝒫 𝐴𝐵))
54rspccv 3606 . . 3 (∀𝑥𝐵 𝒫 𝑥𝐵 → (𝐴𝐵 → 𝒫 𝐴𝐵))
62, 5syl 17 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 → 𝒫 𝐴𝐵))
7 simpl 486 . . . 4 (( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → 𝑥𝐵)
87ralimi 3155 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ∀𝑥𝐵 𝑥𝐵)
9 unieq 4835 . . . . . 6 (𝑥 = 𝒫 𝐴 𝑥 = 𝒫 𝐴)
10 unipw 5330 . . . . . 6 𝒫 𝐴 = 𝐴
119, 10syl6eq 2875 . . . . 5 (𝑥 = 𝒫 𝐴 𝑥 = 𝐴)
1211eleq1d 2900 . . . 4 (𝑥 = 𝒫 𝐴 → ( 𝑥𝐵𝐴𝐵))
1312rspccv 3606 . . 3 (∀𝑥𝐵 𝑥𝐵 → (𝒫 𝐴𝐵𝐴𝐵))
148, 13syl 17 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝒫 𝐴𝐵𝐴𝐵))
156, 14impbid 215 1 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  𝒫 cpw 4522   cuni 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-pw 4524  df-sn 4551  df-pr 4553  df-uni 4825
This theorem is referenced by:  pwinfig  40176
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