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Theorem pwelg 36781
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 475 . . . 4 (( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → 𝒫 𝑥𝐵)
21ralimi 2840 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ∀𝑥𝐵 𝒫 𝑥𝐵)
3 pweq 4014 . . . . 5 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
43eleq1d 2576 . . . 4 (𝑥 = 𝐴 → (𝒫 𝑥𝐵 ↔ 𝒫 𝐴𝐵))
54rspccv 3183 . . 3 (∀𝑥𝐵 𝒫 𝑥𝐵 → (𝐴𝐵 → 𝒫 𝐴𝐵))
62, 5syl 17 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 → 𝒫 𝐴𝐵))
7 simpl 471 . . . 4 (( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → 𝑥𝐵)
87ralimi 2840 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ∀𝑥𝐵 𝑥𝐵)
9 unieq 4278 . . . . . 6 (𝑥 = 𝒫 𝐴 𝑥 = 𝒫 𝐴)
10 unipw 4743 . . . . . 6 𝒫 𝐴 = 𝐴
119, 10syl6eq 2564 . . . . 5 (𝑥 = 𝒫 𝐴 𝑥 = 𝐴)
1211eleq1d 2576 . . . 4 (𝑥 = 𝒫 𝐴 → ( 𝑥𝐵𝐴𝐵))
1312rspccv 3183 . . 3 (∀𝑥𝐵 𝑥𝐵 → (𝒫 𝐴𝐵𝐴𝐵))
148, 13syl 17 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝒫 𝐴𝐵𝐴𝐵))
156, 14impbid 200 1 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  wral 2800  𝒫 cpw 4011   cuni 4270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-pw 4013  df-sn 4029  df-pr 4031  df-uni 4271
This theorem is referenced by:  pwinfig  36782
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