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Metamath Proof Explorer

Theorem elpwgdedVD

Description: Membership in a power class. Theorem 86 of Suppes p. 47. Derived from elpwg . In form of VD deduction with ph and ps as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded is elpwgdedVD using conventional notation. (Contributed by Alan Sare, 23-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses elpwgdedVD.1 ⊢ φ → A ∈ V
elpwgdedVD.2 ⊢ ψ → A ⊆ B
Assertion elpwgdedVD ⊢ φ ψ → A ∈ 𝒫 B

Proof

Step Hyp Ref Expression
1 elpwgdedVD.1 ⊢ φ → A ∈ V
2 elpwgdedVD.2 ⊢ ψ → A ⊆ B
3 elpwg ⊢ A ∈ V → A ∈ 𝒫 B ↔ A ⊆ B
4 3 biimpar ⊢ A ∈ V ∧ A ⊆ B → A ∈ 𝒫 B
5 1 2 4 el12 ⊢ φ ψ → A ∈ 𝒫 B