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Theorem pwuninel 6894
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 6893. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
pwuninel  |-  -.  ~P U. A  e.  A

Proof of Theorem pwuninel
StepHypRef Expression
1 elex 3077 . . . 4  |-  ( ~P
U. A  e.  A  ->  ~P U. A  e. 
_V )
2 pwexb 6487 . . . 4  |-  ( U. A  e.  _V  <->  ~P U. A  e.  _V )
31, 2sylibr 212 . . 3  |-  ( ~P
U. A  e.  A  ->  U. A  e.  _V )
4 pwuninel2 6893 . . 3  |-  ( U. A  e.  _V  ->  -. 
~P U. A  e.  A
)
53, 4syl 16 . 2  |-  ( ~P
U. A  e.  A  ->  -.  ~P U. A  e.  A )
6 id 22 . 2  |-  ( -. 
~P U. A  e.  A  ->  -.  ~P U. A  e.  A )
75, 6pm2.61i 164 1  |-  -.  ~P U. A  e.  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1758   _Vcvv 3068   ~Pcpw 3958   U.cuni 4189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-pw 3960  df-sn 3976  df-pr 3978  df-uni 4190
This theorem is referenced by:  undefnel2  6896  disjen  7568  pnfnre  9526  kelac2lem  29555  kelac2  29556
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