![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 30115). We will later introduce the Axiom of Power Sets ax-pow 5363, which can be expressed in class notation per pwexg 5376. Still later we will prove, in hashpw 14403, 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 4602 | . 2 class 𝒫 𝐴 |
3 | vx | . . . . 5 setvar 𝑥 | |
4 | 3 | cv 1539 | . . . 4 class 𝑥 |
5 | 4, 1 | wss 3948 | . . 3 wff 𝑥 ⊆ 𝐴 |
6 | 5, 3 | cab 2708 | . 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 4605 pweqALT 4617 nfpw 4621 pw0 4815 pwpw0 4816 pwsn 4900 pwsnOLD 4901 vpwex 5375 abssexg 5380 orduniss2 7825 mapex 8832 ssenen 9157 domtriomlem 10443 npex 10987 ustval 24027 avril1 30149 fineqvpow 34560 dfon2lem2 35226 bj-velpwALT 36398 |
Copyright terms: Public domain | W3C validator |