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Metamath Proof Explorer

Theorem elpw2g

Description: Membership in a power class. Theorem 86 of Suppes p. 47. (Contributed by NM, 7-Aug-2000)

Ref Expression
Assertion elpw2g ⊢ B ∈ V → A ∈ 𝒫 B ↔ A ⊆ B

Proof

Step Hyp Ref Expression
1 elpwi ⊢ A ∈ 𝒫 B → A ⊆ B
2 ssexg ⊢ A ⊆ B ∧ B ∈ V → A ∈ V
3 elpwg ⊢ A ∈ V → A ∈ 𝒫 B ↔ A ⊆ B
4 3 biimparc ⊢ A ⊆ B ∧ A ∈ V → A ∈ 𝒫 B
5 2 4 syldan ⊢ A ⊆ B ∧ B ∈ V → A ∈ 𝒫 B
6 5 expcom ⊢ B ∈ V → A ⊆ B → A ∈ 𝒫 B
7 1 6 impbid2 ⊢ B ∈ V → A ∈ 𝒫 B ↔ A ⊆ B