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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 27843). We will later introduce the Axiom of Power Sets ax-pow 5064, which can be expressed in class notation per pwexg 5077. Still later we will prove, in hashpw 13511, 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 4377 | . 2 class 𝒫 𝐴 |
3 | vx | . . . . 5 setvar 𝑥 | |
4 | 3 | cv 1657 | . . . 4 class 𝑥 |
5 | 4, 1 | wss 3797 | . . 3 wff 𝑥 ⊆ 𝐴 |
6 | 5, 3 | cab 2810 | . 2 class {𝑥 ∣ 𝑥 ⊆ 𝐴} |
7 | 2, 6 | wceq 1658 | 1 wff 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} |
Colors of variables: wff setvar class |
This definition is referenced by: pweq 4380 elpw 4383 nfpw 4391 pw0 4560 pwpw0 4561 pwsn 4649 pwsnALT 4650 vpwex 5076 abssexg 5080 orduniss2 7293 mapex 8127 ssenen 8402 domtriomlem 9578 npex 10122 ustval 22375 avril1 27876 dfon2lem2 32226 |
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