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Theorem pwexr 7784
Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5371. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
pwexr (𝒫 𝐴𝑉𝐴 ∈ V)

Proof of Theorem pwexr
StepHypRef Expression
1 unipw 5461 . 2 𝒫 𝐴 = 𝐴
2 uniexg 7759 . 2 (𝒫 𝐴𝑉 𝒫 𝐴 ∈ V)
31, 2eqeltrrid 2844 1 (𝒫 𝐴𝑉𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3478  𝒫 cpw 4605   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913
This theorem is referenced by:  pwexb  7785  pwuninel  8299  pwwf  9845  r1pw  9883  isfin3  10334  dis2ndc  23484  numufl  23939  bj-discrmoore  37094
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