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Theorem pwexr 7785
Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5365. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
pwexr (𝒫 𝐴𝑉𝐴 ∈ V)

Proof of Theorem pwexr
StepHypRef Expression
1 unipw 5455 . 2 𝒫 𝐴 = 𝐴
2 uniexg 7760 . 2 (𝒫 𝐴𝑉 𝒫 𝐴 ∈ V)
31, 2eqeltrrid 2846 1 (𝒫 𝐴𝑉𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Vcvv 3480  𝒫 cpw 4600   cuni 4907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-pw 4602  df-sn 4627  df-pr 4629  df-uni 4908
This theorem is referenced by:  pwexb  7786  pwuninel  8300  pwwf  9847  r1pw  9885  isfin3  10336  dis2ndc  23468  numufl  23923  bj-discrmoore  37112
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