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Mirrors > Home > MPE Home > Th. List > pwel | Structured version Visualization version GIF version |
Description: Quantitative version of pwexg 5372: 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 5300 and ax-pr 5423 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 5372 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V) | |
2 | elssuni 4935 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) | |
3 | 2 | sspwd 4611 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 ∪ 𝐵) |
4 | 1, 3 | elpwd 4604 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3469 𝒫 cpw 4598 ∪ cuni 4903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-pow 5359 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-v 3471 df-in 3951 df-ss 3961 df-pw 4600 df-uni 4904 |
This theorem is referenced by: bj-unirel 36466 |
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