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Theorem pwelg 38826
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 479 . . . 4 (( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → 𝒫 𝑥𝐵)
21ralimi 3134 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ∀𝑥𝐵 𝒫 𝑥𝐵)
3 pweq 4382 . . . . 5 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
43eleq1d 2844 . . . 4 (𝑥 = 𝐴 → (𝒫 𝑥𝐵 ↔ 𝒫 𝐴𝐵))
54rspccv 3508 . . 3 (∀𝑥𝐵 𝒫 𝑥𝐵 → (𝐴𝐵 → 𝒫 𝐴𝐵))
62, 5syl 17 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 → 𝒫 𝐴𝐵))
7 simpl 476 . . . 4 (( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → 𝑥𝐵)
87ralimi 3134 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ∀𝑥𝐵 𝑥𝐵)
9 unieq 4679 . . . . . 6 (𝑥 = 𝒫 𝐴 𝑥 = 𝒫 𝐴)
10 unipw 5150 . . . . . 6 𝒫 𝐴 = 𝐴
119, 10syl6eq 2830 . . . . 5 (𝑥 = 𝒫 𝐴 𝑥 = 𝐴)
1211eleq1d 2844 . . . 4 (𝑥 = 𝒫 𝐴 → ( 𝑥𝐵𝐴𝐵))
1312rspccv 3508 . . 3 (∀𝑥𝐵 𝑥𝐵 → (𝒫 𝐴𝐵𝐴𝐵))
148, 13syl 17 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝒫 𝐴𝐵𝐴𝐵))
156, 14impbid 204 1 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wral 3090  𝒫 cpw 4379   cuni 4671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-pw 4381  df-sn 4399  df-pr 4401  df-uni 4672
This theorem is referenced by:  pwinfig  38827
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