MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwsn Structured version   Unicode version

Theorem pwsn 4180
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 4126 . . 3  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
21abbii 2583 . 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 2489 1  |-  ~P { A }  =  { (/)
,  { A } }
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    = wceq 1370   {cab 2436    C_ wss 3423   (/)c0 3732   ~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 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-v 3067  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-pw 3957  df-sn 3973  df-pr 3975
This theorem is referenced by:  pmtrsn  16124  topsn  18653  concompid  19148  usgra1v  23440  esumsn  26646  cvmlift2lem9  27331
  Copyright terms: Public domain W3C validator