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Theorem pwuninel 7931
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 7930. (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 7477 . . 3 (𝒫 𝐴𝐴 𝐴 ∈ V)
2 pwuninel2 7930 . . 3 ( 𝐴 ∈ V → ¬ 𝒫 𝐴𝐴)
31, 2syl 17 . 2 (𝒫 𝐴𝐴 → ¬ 𝒫 𝐴𝐴)
4 id 22 . 2 (¬ 𝒫 𝐴𝐴 → ¬ 𝒫 𝐴𝐴)
53, 4pm2.61i 185 1 ¬ 𝒫 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2115  Vcvv 3480  𝒫 cpw 4521   cuni 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-pw 4523  df-sn 4550  df-pr 4552  df-uni 4825
This theorem is referenced by:  undefnel2  7933  disjen  8665  pnfnre  10674  kelac2lem  39861  kelac2  39862  ndfatafv2nrn  43640  afv2ndefb  43643
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