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Mirrors > Home > ILE Home > Th. List > pwpwab | 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 2742 | . . 3 ⊢ 𝑥 ∈ V | |
2 | elpwpw 3975 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ ∪ 𝑥 ⊆ 𝐴)) | |
3 | 1, 2 | mpbiran 940 | . 2 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴) |
4 | 3 | abbi2i 2292 | 1 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 {cab 2163 Vcvv 2739 ⊆ wss 3131 𝒫 cpw 3577 ∪ cuni 3811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2741 df-in 3137 df-ss 3144 df-pw 3579 df-uni 3812 |
This theorem is referenced by: pwpwssunieq 3977 |
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