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Theorem pwexr 7744
Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5323. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
pwexr (𝒫 𝐴𝑉𝐴 ∈ V)

Proof of Theorem pwexr
StepHypRef Expression
1 unipw 5413 . 2 𝒫 𝐴 = 𝐴
2 uniexg 7719 . 2 (𝒫 𝐴𝑉 𝒫 𝐴 ∈ V)
31, 2eqeltrrid 2834 1 (𝒫 𝐴𝑉𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2109  Vcvv 3450  𝒫 cpw 4566   cuni 4874
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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-pr 5390  ax-un 7714
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-un 3922  df-ss 3934  df-pw 4568  df-sn 4593  df-pr 4595  df-uni 4875
This theorem is referenced by:  pwexb  7745  pwuninel  8257  pwwf  9767  r1pw  9805  isfin3  10256  dis2ndc  23354  numufl  23809  bj-discrmoore  37106
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