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| Description: Quantitative version of pwexg 5377: the powerset of an element of a class is an element of the double powerclass of the union of that class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) Remove use of ax-nul 5305 and ax-pr 5431 and shorten proof. (Revised by BJ, 13-Apr-2024.) | 
| Ref | Expression | 
|---|---|
| pwel | ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pwexg 5377 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V) | |
| 2 | elssuni 4936 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) | |
| 3 | 2 | sspwd 4612 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵) | 
| 4 | 1, 3 | elpwd 4605 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3479 𝒫 cpw 4599 ∪ cuni 4906 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-pow 5364 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-pw 4601 df-uni 4907 | 
| This theorem is referenced by: bj-unirel 37053 | 
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