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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 3470 | . . 3 ⊢ 𝑥 ∈ V | |
2 | elpwpw 5096 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ ∪ 𝑥 ⊆ 𝐴)) | |
3 | 1, 2 | mpbiran 706 | . 2 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴) |
4 | 3 | eqabi 2861 | 1 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {cab 2701 Vcvv 3466 ⊆ wss 3941 𝒫 cpw 4595 ∪ cuni 4900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-v 3468 df-in 3948 df-ss 3958 df-pw 4597 df-uni 4901 |
This theorem is referenced by: pwpwssunieq 5098 |
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