Theorem elpwgdedVD 44915

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 4608. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 44562 is elpwgdedVD 44915 using conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elpwgdedVD.1	⊢ (   𝜑   ▶   𝐴 ∈ V   )
elpwgdedVD.2	⊢ (   𝜓   ▶   𝐴 ⊆ 𝐵   )

Assertion

Ref	Expression
elpwgdedVD	⊢ (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )

Proof of Theorem elpwgdedVD

Step	Hyp	Ref	Expression
1		elpwgdedVD.1	. 2 ⊢ (   𝜑   ▶   𝐴 ∈ V   )
2		elpwgdedVD.2	. 2 ⊢ (   𝜓   ▶   𝐴 ⊆ 𝐵   )
3		elpwg 4608	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
4	3	biimpar 477	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵)
5	1, 2, 4	el12 44724	1 ⊢ (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )

Syntax hints:   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605  (   wvd1 44567  (   wvhc2 44578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ss 3980  df-pw 4607  df-vd1 44568  df-vhc2 44579

This theorem is referenced by:  sspwimpVD  44917