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Theorem pwuninel 6545
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 6544. (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 2964 . . . 4  |-  ( ~P
U. A  e.  A  ->  ~P U. A  e. 
_V )
2 pwexb 4753 . . . 4  |-  ( U. A  e.  _V  <->  ~P U. A  e.  _V )
31, 2sylibr 204 . . 3  |-  ( ~P
U. A  e.  A  ->  U. A  e.  _V )
4 pwuninel2 6544 . . 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 20 . 2  |-  ( -. 
~P U. A  e.  A  ->  -.  ~P U. A  e.  A )
75, 6pm2.61i 158 1  |-  -.  ~P U. A  e.  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1725   _Vcvv 2956   ~Pcpw 3799   U.cuni 4015
This theorem is referenced by:  undefnel2  6547  disjen  7264  pnfnre  9127  kelac2lem  27139  kelac2  27140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-pw 3801  df-sn 3820  df-pr 3821  df-uni 4016
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