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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-unirel | Structured version Visualization version GIF version |
Description: Quantitative version of uniexr 7478: 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 4868 | . . 3 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴 | |
2 | pwel 5275 | . . 3 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉) | |
3 | bj-sselpwuni 34367 | . . 3 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉) | |
4 | 1, 2, 3 | sylancr 589 | . 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉) |
5 | unipw 5336 | . . 3 ⊢ ∪ 𝒫 𝒫 ∪ 𝑉 = 𝒫 ∪ 𝑉 | |
6 | 5 | pweqi 4550 | . 2 ⊢ 𝒫 ∪ 𝒫 𝒫 ∪ 𝑉 = 𝒫 𝒫 ∪ 𝑉 |
7 | 4, 6 | eleqtrdi 2922 | 1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ⊆ wss 3929 𝒫 cpw 4532 ∪ cuni 4831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-pw 4534 df-sn 4561 df-pr 4563 df-uni 4832 |
This theorem is referenced by: (None) |
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