Theorem elpwgdedVD 44941

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 4578. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 44589 is elpwgdedVD 44941 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 4578	. . 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 44750	1 ⊢ (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )

Syntax hints:   ∈ wcel 2108  Vcvv 3459   ⊆ wss 3926  𝒫 cpw 4575  (   wvd1 44594  (   wvhc2 44605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ss 3943  df-pw 4577  df-vd1 44595  df-vhc2 44606

This theorem is referenced by:  sspwimpVD  44943