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Theorem pwexr 7208
Description: Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5036. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
pwexr (𝒫 𝐴𝑉𝐴 ∈ V)

Proof of Theorem pwexr
StepHypRef Expression
1 unipw 5110 . 2 𝒫 𝐴 = 𝐴
2 uniexg 7190 . 2 (𝒫 𝐴𝑉 𝒫 𝐴 ∈ V)
31, 2syl5eqelr 2884 1 (𝒫 𝐴𝑉𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  Vcvv 3386  𝒫 cpw 4350   cuni 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-rex 3096  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-pw 4352  df-sn 4370  df-pr 4372  df-uni 4630
This theorem is referenced by:  pwexb  7209  pwuninel  7640  pwfi  8504  pwwf  8921  r1pw  8959  isfin3  9407  dis2ndc  21591  numufl  22046  bj-discrmoore  33558
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