Theorem elpwg

Description: Membership in a power class. Theorem 86 of Suppes p. 47. See also elpw2g . (Contributed by NM, 6-Aug-2000) (Proof shortened by BJ, 31-Dec-2023)

	Ref	Expression
Assertion	elpwg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sseq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵 ) )
2		sseq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )
3		df-pw	⊢ 𝒫 𝐵 = { 𝑥 ∣ 𝑥 ⊆ 𝐵 }
4	1 2 3	elab2gw	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵 ) )