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Theorem pwuninel 8079
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 8078. (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 7606 . . 3 (𝒫 𝐴𝐴 𝐴 ∈ V)
2 pwuninel2 8078 . . 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 2106  Vcvv 3430  𝒫 cpw 4534   cuni 4840
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-pw 4536  df-sn 4563  df-pr 4565  df-uni 4841
This theorem is referenced by:  undefnel2  8081  disjen  8909  pnfnre  11004  kelac2lem  40875  kelac2  40876  ndfatafv2nrn  44669  afv2ndefb  44672
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