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Theorem elpwi2 4978
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 4976 . . 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 2139  wss 3715  𝒫 cpw 4302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304
This theorem is referenced by:  sprsymrelfolem1  42252
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