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Theorem bj-unirel 34463
Description: Quantitative version of uniexr 7469: if the union of a class is an element of a class, then that class is an element of the double powerclass of the union of this class. (Contributed by BJ, 6-Apr-2024.)
Assertion
Ref Expression
bj-unirel ( 𝐴𝑉𝐴 ∈ 𝒫 𝒫 𝑉)

Proof of Theorem bj-unirel
StepHypRef Expression
1 pwuni 4840 . . 3 𝐴 ⊆ 𝒫 𝐴
2 pwel 5250 . . 3 ( 𝐴𝑉 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝑉)
3 bj-sselpwuni 34462 . . 3 ((𝐴 ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ 𝒫 𝒫 𝑉) → 𝐴 ∈ 𝒫 𝒫 𝒫 𝑉)
41, 2, 3sylancr 590 . 2 ( 𝐴𝑉𝐴 ∈ 𝒫 𝒫 𝒫 𝑉)
5 unipw 5311 . . 3 𝒫 𝒫 𝑉 = 𝒫 𝑉
65pweqi 4518 . 2 𝒫 𝒫 𝒫 𝑉 = 𝒫 𝒫 𝑉
74, 6eleqtrdi 2903 1 ( 𝐴𝑉𝐴 ∈ 𝒫 𝒫 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  wss 3884  𝒫 cpw 4500   cuni 4803
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
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 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531  df-uni 4804
This theorem is referenced by: (None)
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