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Theorem pwpr 3839
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 3793 . . . 4  |-  ( x 
C_  { A ,  B }  <->  ( ( x  =  (/)  \/  x  =  { A } )  \/  ( x  =  { B }  \/  x  =  { A ,  B } ) ) )
2 vex 2804 . . . . . 6  |-  x  e. 
_V
32elpr 3671 . . . . 5  |-  ( x  e.  { (/) ,  { A } }  <->  ( x  =  (/)  \/  x  =  { A } ) )
42elpr 3671 . . . . 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 3644 . . 3  |-  ( x  e.  ~P { A ,  B }  <->  x  C_  { A ,  B } )
8 elun 3329 . . 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 2293 1  |-  ~P { A ,  B }  =  ( { (/) ,  { A } }  u.  { { B } ,  { A ,  B } } )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    = wceq 1632    e. wcel 1696    u. cun 3163    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   {cpr 3654
This theorem is referenced by:  pwpwpw0  3841  ord3ex  4216  prsiga  23507
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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-ral 2561  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
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