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Theorem pwuninel 8207
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 8206. (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 7700 . . 3 (𝒫 𝐴𝐴 𝐴 ∈ V)
2 pwuninel2 8206 . . 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 2107  Vcvv 3444  𝒫 cpw 4561   cuni 4866
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-un 3916  df-in 3918  df-ss 3928  df-pw 4563  df-sn 4588  df-pr 4590  df-uni 4867
This theorem is referenced by:  undefnel2  8209  disjen  9081  pnfnre  11201  kelac2lem  41434  kelac2  41435  ndfatafv2nrn  45539  afv2ndefb  45542
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