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Theorem pwpr 4011
Description: The power set of an unordered pair. (Contributed by NM, 1-May-2009.)
Assertion
Ref Expression
pwpr  |-  ~P { A ,  B }  =  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )

Proof of Theorem pwpr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sspr 3962 . . . 4  |-  ( x 
C_  { A ,  B }  <->  ( ( x  =  (/)  \/  x  =  { A } )  \/  ( x  =  { B }  \/  x  =  { A ,  B } ) ) )
2 vex 2959 . . . . . 6  |-  x  e. 
_V
32elpr 3832 . . . . 5  |-  ( x  e.  { (/) ,  { A } }  <->  ( x  =  (/)  \/  x  =  { A } ) )
42elpr 3832 . . . . 5  |-  ( x  e.  { { B } ,  { A ,  B } }  <->  ( x  =  { B }  \/  x  =  { A ,  B } ) )
53, 4orbi12i 508 . . . 4  |-  ( ( x  e.  { (/) ,  { A } }  \/  x  e.  { { B } ,  { A ,  B } } )  <-> 
( ( x  =  (/)  \/  x  =  { A } )  \/  (
x  =  { B }  \/  x  =  { A ,  B }
) ) )
61, 5bitr4i 244 . . 3  |-  ( x 
C_  { A ,  B }  <->  ( x  e. 
{ (/) ,  { A } }  \/  x  e.  { { B } ,  { A ,  B } } ) )
72elpw 3805 . . 3  |-  ( x  e.  ~P { A ,  B }  <->  x  C_  { A ,  B } )
8 elun 3488 . . 3  |-  ( x  e.  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )  <->  ( x  e.  { (/) ,  { A } }  \/  x  e.  { { B } ,  { A ,  B } } ) )
96, 7, 83bitr4i 269 . 2  |-  ( x  e.  ~P { A ,  B }  <->  x  e.  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } ) )
109eqriv 2433 1  |-  ~P { A ,  B }  =  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )
Colors of variables: wff set class
Syntax hints:    \/ wo 358    = wceq 1652    e. wcel 1725    u. cun 3318    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   {cpr 3815
This theorem is referenced by:  pwpwpw0  4013  ord3ex  4389  prsiga  24514
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
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-ral 2710  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
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