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Theorem elpwi2 4970
 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 4968 . . 3 (𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
42, 3ax-mp 5 . 2 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
51, 4mpbir 221 1 𝐴 ∈ 𝒫 𝐵
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∈ wcel 2131   ⊆ wss 3707  𝒫 cpw 4294 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-v 3334  df-in 3714  df-ss 3721  df-pw 4296 This theorem is referenced by:  sprsymrelfolem1  42244
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