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| Mirrors > Home > MPE Home > Th. List > pwpwab | Structured version Visualization version GIF version | ||
| Description: The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.) | 
| Ref | Expression | 
|---|---|
| pwpwab | ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vex 3484 | . . 3 ⊢ 𝑥 ∈ V | |
| 2 | elpwpw 5102 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ ∪ 𝑥 ⊆ 𝐴)) | |
| 3 | 1, 2 | mpbiran 709 | . 2 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴) | 
| 4 | 3 | eqabi 2877 | 1 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-ss 3968 df-pw 4602 df-uni 4908 | 
| This theorem is referenced by: pwpwssunieq 5104 | 
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