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Metamath Proof Explorer

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 = A → x ⊆ B ↔ A ⊆ B
2 df-pw ⊢ 𝒫 B = x | x ⊆ B
3 1 2 elab2g ⊢ A ∈ V → A ∈ 𝒫 B ↔ A ⊆ B