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Theorem elpwbi 39430
 Description: Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.)
Hypothesis
Ref Expression
elpwbi.1 𝐵 ∈ V
Assertion
Ref Expression
elpwbi (𝐴𝐵𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwbi
StepHypRef Expression
1 elpwbi.1 . . 3 𝐵 ∈ V
21elpw2 5213 . 2 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
32bicomi 227 1 (𝐴𝐵𝐴 ∈ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-sep 5168 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499 This theorem is referenced by: (None)
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