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Theorem elpwgdedVD 38072
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 4019. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 37698 is elpwgdedVD 38072 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   ▶   𝐴𝐵   )
Assertion
Ref Expression
elpwgdedVD (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )

Proof of Theorem elpwgdedVD
StepHypRef Expression
1 elpwgdedVD.1 . 2 (   𝜑   ▶   𝐴 ∈ V   )
2 elpwgdedVD.2 . 2 (   𝜓   ▶   𝐴𝐵   )
3 elpwg 4019 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
43biimpar 500 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) → 𝐴 ∈ 𝒫 𝐵)
51, 2, 4el12 37871 1 (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )
Colors of variables: wff setvar class
Syntax hints:  wcel 1938  Vcvv 3077  wss 3444  𝒫 cpw 4011  (   wvd1 37703  (   wvhc2 37714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-v 3079  df-in 3451  df-ss 3458  df-pw 4013  df-vd1 37704  df-vhc2 37715
This theorem is referenced by:  sspwimpVD  38074
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