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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-unirel | Structured version Visualization version GIF version |
Description: Quantitative version of uniexr 7526: 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.) |
Ref | Expression |
---|---|
bj-unirel | ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwuni 4844 | . . 3 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
2 | pwel 5259 | . . 3 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉) | |
3 | bj-sselpwuni 34909 | . . 3 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉) | |
4 | 1, 2, 3 | sylancr 590 | . 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉) |
5 | unipw 5320 | . . 3 ⊢ ∪ 𝒫 𝒫 ∪ 𝑉 = 𝒫 ∪ 𝑉 | |
6 | 5 | pweqi 4517 | . 2 ⊢ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉 = 𝒫 𝒫 ∪ 𝑉 |
7 | 4, 6 | eleqtrdi 2841 | 1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ⊆ wss 3853 𝒫 cpw 4499 ∪ cuni 4805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-pw 4501 df-sn 4528 df-pr 4530 df-uni 4806 |
This theorem is referenced by: (None) |
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