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Metamath Proof Explorer

Theorem elpw

Description: Membership in a power class. Theorem 86 of Suppes p. 47. (Contributed by NM, 31-Dec-1993) (Proof shortened by BJ, 31-Dec-2023)

Ref Expression
Hypothesis elpw.1 ⊢ A ∈ V
Assertion elpw ⊢ A ∈ 𝒫 B ↔ A ⊆ B

Proof

Step Hyp Ref Expression
1 elpw.1 ⊢ A ∈ V
2 elpwg ⊢ A ∈ V → A ∈ 𝒫 B ↔ A ⊆ B
3 1 2 ax-mp ⊢ A ∈ 𝒫 B ↔ A ⊆ B