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Theorem pwsn 4239
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 4185 . . 3  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
21abbii 2601 . 2  |-  { x  |  x  C_  { A } }  =  {
x  |  ( x  =  (/)  \/  x  =  { A } ) }
3 df-pw 4012 . 2  |-  ~P { A }  =  {
x  |  x  C_  { A } }
4 dfpr2 4042 . 2  |-  { (/) ,  { A } }  =  { x  |  ( x  =  (/)  \/  x  =  { A } ) }
52, 3, 43eqtr4i 2506 1  |-  ~P { A }  =  { (/)
,  { A } }
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    = wceq 1379   {cab 2452    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   {cpr 4029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-pw 4012  df-sn 4028  df-pr 4030
This theorem is referenced by:  pmtrsn  16337  topsn  19200  concompid  19695  usgra1v  24063  esumsn  27709  cvmlift2lem9  28393
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