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Theorem pwexr 7767
Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5345. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
pwexr (𝒫 𝐴𝑉𝐴 ∈ V)

Proof of Theorem pwexr
StepHypRef Expression
1 unipw 5435 . 2 𝒫 𝐴 = 𝐴
2 uniexg 7742 . 2 (𝒫 𝐴𝑉 𝒫 𝐴 ∈ V)
31, 2eqeltrrid 2838 1 (𝒫 𝐴𝑉𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3463  𝒫 cpw 4580   cuni 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-pr 5412  ax-un 7737
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-un 3936  df-ss 3948  df-pw 4582  df-sn 4607  df-pr 4609  df-uni 4888
This theorem is referenced by:  pwexb  7768  pwuninel  8282  pwwf  9829  r1pw  9867  isfin3  10318  dis2ndc  23414  numufl  23869  bj-discrmoore  37071
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