Theorem elpwgdedVD 45645

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 4565. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 45293 is elpwgdedVD 45645 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 4565	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
4	3	biimpar 482	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ 𝒫 𝐵)
5	1, 2, 4	el12 45454	1 ⊢ (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )

Syntax hints:   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562  (   wvd1 45298  (   wvhc2 45309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ss 3922  df-pw 4564  df-vd1 45299  df-vhc2 45310

This theorem is referenced by:  sspwimpVD  45647