![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2644 | . . 3 ⊢ 𝑥 ∈ V | |
2 | elpwpw 3845 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ ∪ 𝑥 ⊆ 𝐴)) | |
3 | 1, 2 | mpbiran 892 | . 2 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴) |
4 | 3 | abbi2i 2214 | 1 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} |
Colors of variables: wff set class |
Syntax hints: = wceq 1299 ∈ wcel 1448 {cab 2086 Vcvv 2641 ⊆ wss 3021 𝒫 cpw 3457 ∪ cuni 3683 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-v 2643 df-in 3027 df-ss 3034 df-pw 3459 df-uni 3684 |
This theorem is referenced by: pwpwssunieq 3847 |
Copyright terms: Public domain | W3C validator |