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Theorem pwsn 4185
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 4130 . . 3  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
21abbii 2536 . 2  |-  { x  |  x  C_  { A } }  =  {
x  |  ( x  =  (/)  \/  x  =  { A } ) }
3 df-pw 3957 . 2  |-  ~P { A }  =  {
x  |  x  C_  { A } }
4 dfpr2 3987 . 2  |-  { (/) ,  { A } }  =  { x  |  ( x  =  (/)  \/  x  =  { A } ) }
52, 3, 43eqtr4i 2441 1  |-  ~P { A }  =  { (/)
,  { A } }
Colors of variables: wff setvar class
Syntax hints:    \/ wo 366    = wceq 1405   {cab 2387    C_ wss 3414   (/)c0 3738   ~Pcpw 3955   {csn 3972   {cpr 3974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-pw 3957  df-sn 3973  df-pr 3975
This theorem is referenced by:  pmtrsn  16868  topsn  19728  concompid  20224  usgra1v  24807  esumsnf  28511  cvmlift2lem9  29608
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