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Theorem pwuninel 7001
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 7000. (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 3122 . . . 4  |-  ( ~P
U. A  e.  A  ->  ~P U. A  e. 
_V )
2 pwexb 6589 . . . 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 7000 . . 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 1767   _Vcvv 3113   ~Pcpw 4010   U.cuni 4245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-pw 4012  df-sn 4028  df-pr 4030  df-uni 4246
This theorem is referenced by:  undefnel2  7003  disjen  7671  pnfnre  9631  kelac2lem  30614  kelac2  30615
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