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Metamath Proof Explorer

Definition df-pw

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 ). We will later introduce the Axiom of Power Sets ax-pow , which can be expressed in class notation per pwexg . Still later we will prove, in hashpw , 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
	Assertion	df-pw	⊢ 𝒫 𝐴 = { 𝑥 ∣ 𝑥 ⊆ 𝐴 }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	⊢ 𝐴
1	0	cpw	⊢ 𝒫 𝐴
2		vx	⊢ 𝑥
3	2	cv	⊢ 𝑥
4	3 0	wss	⊢ 𝑥 ⊆ 𝐴
5	4 2	cab	⊢ { 𝑥 ∣ 𝑥 ⊆ 𝐴 }
6	1 5	wceq	⊢ 𝒫 𝐴 = { 𝑥 ∣ 𝑥 ⊆ 𝐴 }