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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 26678). We will later introduce the Axiom of Power Sets ax-pow 4769, which can be expressed in class notation per pwexg 4776. Still later we will prove, in hashpw 13083, 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 4108 | . 2 class 𝒫 𝐴 |
3 | vx | . . . . 5 setvar 𝑥 | |
4 | 3 | cv 1474 | . . . 4 class 𝑥 |
5 | 4, 1 | wss 3540 | . . 3 wff 𝑥 ⊆ 𝐴 |
6 | 5, 3 | cab 2596 | . 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴} |
7 | 2, 6 | wceq 1475 | 1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} |
Colors of variables: wff setvar class |
This definition is referenced by: pweq 4111 elpw 4114 nfpw 4120 pw0 4283 pwpw0 4284 pwsn 4366 pwsnALT 4367 pwex 4774 abssexg 4777 orduniss2 6925 mapex 7750 ssenen 8019 domtriomlem 9147 npex 9687 ustval 21816 avril1 26711 dfon2lem2 30933 |
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