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| Mirrors > Home > MPE Home > Th. List > df-pw | Structured version Visualization version GIF version | ||
| Description: Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if 𝐴 = {3, 5, 7}, then 𝒫 𝐴 = {∅, {3}, {5}, {7}, {3, 5}, {3, 7}, {5, 7}, {3, 5, 7}} (ex-pw 30448). We will later introduce the Axiom of Power Sets ax-pow 5365, which can be expressed in class notation per pwexg 5378. Still later we will prove, in hashpw 14475, that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 24-Jun-1993.) |
| Ref | Expression |
|---|---|
| df-pw | ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cA | . . 3 class 𝐴 | |
| 2 | 1 | cpw 4600 | . 2 class 𝒫 𝐴 |
| 3 | vx | . . . . 5 setvar 𝑥 | |
| 4 | 3 | cv 1539 | . . . 4 class 𝑥 |
| 5 | 4, 1 | wss 3951 | . . 3 wff 𝑥 ⊆ 𝐴 |
| 6 | 5, 3 | cab 2714 | . 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴} |
| 7 | 2, 6 | wceq 1540 | 1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} |
| Colors of variables: wff setvar class |
| This definition is referenced by: elpwg 4603 pweqALT 4615 nfpw 4619 pw0 4812 pwpw0 4813 pwsn 4900 vpwex 5377 abssexg 5382 orduniss2 7853 mapexOLD 8872 ssenen 9191 domtriomlem 10482 npex 11026 ustval 24211 avril1 30482 fineqvpow 35110 dfon2lem2 35785 bj-velpwALT 37054 |
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