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Theorem pwuninel 8277
Description: The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. See also pwuninel2 8276. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
pwuninel ¬ 𝒫 𝐴𝐴

Proof of Theorem pwuninel
StepHypRef Expression
1 pwexr 7763 . . 3 (𝒫 𝐴𝐴 𝐴 ∈ V)
2 pwuninel2 8276 . . 3 ( 𝐴 ∈ V → ¬ 𝒫 𝐴𝐴)
31, 2syl 17 . 2 (𝒫 𝐴𝐴 → ¬ 𝒫 𝐴𝐴)
4 id 22 . 2 (¬ 𝒫 𝐴𝐴 → ¬ 𝒫 𝐴𝐴)
53, 4pm2.61i 182 1 ¬ 𝒫 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2098  Vcvv 3463  𝒫 cpw 4596   cuni 4901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5292  ax-pr 5421  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-un 3944  df-in 3946  df-ss 3956  df-pw 4598  df-sn 4623  df-pr 4625  df-uni 4902
This theorem is referenced by:  undefnel2  8279  disjen  9155  pnfnre  11283  kelac2lem  42525  kelac2  42526  ndfatafv2nrn  46636  afv2ndefb  46639
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