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Theorem elpwi2 38291
 Description: Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
elpwi2.1 𝐵𝑉
elpwi2.2 𝐴𝐵
Assertion
Ref Expression
elpwi2 𝐴 ∈ 𝒫 𝐵

Proof of Theorem elpwi2
StepHypRef Expression
1 elpwi2.2 . 2 𝐴𝐵
2 elpwi2.1 . . 3 𝐵𝑉
3 elpw2g 4754 . . 3 (𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
42, 3ax-mp 5 . 2 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
51, 4mpbir 220 1 𝐴 ∈ 𝒫 𝐵
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977   ⊆ wss 3540  𝒫 cpw 4108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110 This theorem is referenced by: (None)
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