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Theorem elpwgdedVD 39675
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived from elpwg 4306. In form of VD deduction with 𝜑 and 𝜓 as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded 39305 is elpwgdedVD 39675 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 4306 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
43biimpar 463 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) → 𝐴 ∈ 𝒫 𝐵)
51, 2, 4el12 39478 1 (   (   𝜑   ,   𝜓   )   ▶   𝐴 ∈ 𝒫 𝐵   )
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  Vcvv 3351  wss 3723  𝒫 cpw 4298  (   wvd1 39310  (   wvhc2 39321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-ss 3737  df-pw 4300  df-vd1 39311  df-vhc2 39322
This theorem is referenced by:  sspwimpVD  39677
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