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    )