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Theorem pwexr 7479
 Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5257. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
pwexr (𝒫 𝐴𝑉𝐴 ∈ V)

Proof of Theorem pwexr
StepHypRef Expression
1 unipw 5333 . 2 𝒫 𝐴 = 𝐴
2 uniexg 7458 . 2 (𝒫 𝐴𝑉 𝒫 𝐴 ∈ V)
31, 2eqeltrrid 2916 1 (𝒫 𝐴𝑉𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3493  𝒫 cpw 4537  ∪ cuni 4830 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rex 3142  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-pw 4539  df-sn 4560  df-pr 4562  df-uni 4831 This theorem is referenced by:  pwexb  7480  pwuninel  7933  pwfi  8811  pwwf  9228  r1pw  9266  isfin3  9710  dis2ndc  22060  numufl  22515  bj-discrmoore  34395
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