Theorem elpwgdedVD 42097

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 4491. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 41744 is elpwgdedVD 42097 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 4491	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
4	3	biimpar 481	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵)
5	1, 2, 4	el12 41906	1 ⊢ (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )

Syntax hints:   ∈ wcel 2114  Vcvv 3398   ⊆ wss 3843  𝒫 cpw 4488  (   wvd1 41749  (   wvhc2 41760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-in 3850  df-ss 3860  df-pw 4490  df-vd1 41750  df-vhc2 41761

This theorem is referenced by:  sspwimpVD  42099