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Theorem pwexr 7704
Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5325. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
pwexr (𝒫 𝐴𝑉𝐴 ∈ V)

Proof of Theorem pwexr
StepHypRef Expression
1 unipw 5412 . 2 𝒫 𝐴 = 𝐴
2 uniexg 7682 . 2 (𝒫 𝐴𝑉 𝒫 𝐴 ∈ V)
31, 2eqeltrrid 2843 1 (𝒫 𝐴𝑉𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3448  𝒫 cpw 4565   cuni 4870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-un 3920  df-in 3922  df-ss 3932  df-pw 4567  df-sn 4592  df-pr 4594  df-uni 4871
This theorem is referenced by:  pwexb  7705  pwuninel  8211  pwfiOLD  9298  pwwf  9750  r1pw  9788  isfin3  10239  dis2ndc  22827  numufl  23282  bj-discrmoore  35611
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