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Theorem pwsn 3837
Description: The power set of a singleton. (Contributed by NM, 5-Jun-2006.)
Assertion
Ref Expression
pwsn  |-  ~P { A }  =  { (/)
,  { A } }

Proof of Theorem pwsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sssn 3788 . . 3  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
21abbii 2408 . 2  |-  { x  |  x  C_  { A } }  =  {
x  |  ( x  =  (/)  \/  x  =  { A } ) }
3 df-pw 3640 . 2  |-  ~P { A }  =  {
x  |  x  C_  { A } }
4 dfpr2 3669 . 2  |-  { (/) ,  { A } }  =  { x  |  ( x  =  (/)  \/  x  =  { A } ) }
52, 3, 43eqtr4i 2326 1  |-  ~P { A }  =  { (/)
,  { A } }
Colors of variables: wff set class
Syntax hints:    \/ wo 357    = wceq 1632   {cab 2282    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   {cpr 3654
This theorem is referenced by:  topsn  16689  concompid  17173  esumsn  23452  cvmlift2lem9  23857  usgra1v  28260
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-ne 2461  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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