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Metamath Proof Explorer

Theorem elpw2

Description: Membership in a power class. Theorem 86 of Suppes p. 47. (Contributed by NM, 11-Oct-2007)

Ref Expression
Hypothesis elpw2.1 ⊢ B ∈ V
Assertion elpw2 ⊢ A ∈ 𝒫 B ↔ A ⊆ B

Proof

Step Hyp Ref Expression
1 elpw2.1 ⊢ B ∈ V
2 elpw2g ⊢ B ∈ V → A ∈ 𝒫 B ↔ A ⊆ B
3 1 2 ax-mp ⊢ A ∈ 𝒫 B ↔ A ⊆ B