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Theorem pwuninel 6316
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 6315. (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 2809 . . . 4  |-  ( ~P
U. A  e.  A  ->  ~P U. A  e. 
_V )
2 pwexb 4580 . . . 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 6315 . . 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 1696   _Vcvv 2801   ~Pcpw 3638   U.cuni 3843
This theorem is referenced by:  undefnel2  6318  disjen  7034  pnfnre  8890  kelac2lem  27265  kelac2  27266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660  df-uni 3844
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