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Theorem pwpr 3823
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 3777 . . . 4  |-  ( x 
C_  { A ,  B }  <->  ( ( x  =  (/)  \/  x  =  { A } )  \/  ( x  =  { B }  \/  x  =  { A ,  B } ) ) )
2 vex 2791 . . . . . 6  |-  x  e. 
_V
32elpr 3658 . . . . 5  |-  ( x  e.  { (/) ,  { A } }  <->  ( x  =  (/)  \/  x  =  { A } ) )
42elpr 3658 . . . . 5  |-  ( x  e.  { { B } ,  { A ,  B } }  <->  ( x  =  { B }  \/  x  =  { A ,  B } ) )
53, 4orbi12i 507 . . . 4  |-  ( ( x  e.  { (/) ,  { A } }  \/  x  e.  { { B } ,  { A ,  B } } )  <-> 
( ( x  =  (/)  \/  x  =  { A } )  \/  (
x  =  { B }  \/  x  =  { A ,  B }
) ) )
61, 5bitr4i 243 . . 3  |-  ( x 
C_  { A ,  B }  <->  ( x  e. 
{ (/) ,  { A } }  \/  x  e.  { { B } ,  { A ,  B } } ) )
72elpw 3631 . . 3  |-  ( x  e.  ~P { A ,  B }  <->  x  C_  { A ,  B } )
8 elun 3316 . . 3  |-  ( x  e.  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )  <->  ( x  e.  { (/) ,  { A } }  \/  x  e.  { { B } ,  { A ,  B } } ) )
96, 7, 83bitr4i 268 . 2  |-  ( x  e.  ~P { A ,  B }  <->  x  e.  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } ) )
109eqriv 2280 1  |-  ~P { A ,  B }  =  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    = wceq 1623    e. wcel 1684    u. cun 3150    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   {cpr 3641
This theorem is referenced by:  pwpwpw0  3825  ord3ex  4200  prsiga  23492
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-ral 2548  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
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