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Theorem pwuninel 6300
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 6299. (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 2796 . . . 4  |-  ( ~P
U. A  e.  A  ->  ~P U. A  e. 
_V )
2 pwexb 4564 . . . 4  |-  ( U. A  e.  _V  <->  ~P U. A  e.  _V )
31, 2sylibr 203 . . 3  |-  ( ~P
U. A  e.  A  ->  U. A  e.  _V )
4 pwuninel2 6299 . . 3  |-  ( U. A  e.  _V  ->  -. 
~P U. A  e.  A
)
53, 4syl 15 . 2  |-  ( ~P
U. A  e.  A  ->  -.  ~P U. A  e.  A )
6 id 19 . 2  |-  ( -. 
~P U. A  e.  A  ->  -.  ~P U. A  e.  A )
75, 6pm2.61i 156 1  |-  -.  ~P U. A  e.  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   U.cuni 3827
This theorem is referenced by:  undefnel2  6302  disjen  7018  pnfnre  8874  kelac2lem  27162  kelac2  27163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828
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