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

Theorem pwsn 4009
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 3957 . . 3  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
21abbii 2548 . 2  |-  { x  |  x  C_  { A } }  =  {
x  |  ( x  =  (/)  \/  x  =  { A } ) }
3 df-pw 3801 . 2  |-  ~P { A }  =  {
x  |  x  C_  { A } }
4 dfpr2 3830 . 2  |-  { (/) ,  { A } }  =  { x  |  ( x  =  (/)  \/  x  =  { A } ) }
52, 3, 43eqtr4i 2466 1  |-  ~P { A }  =  { (/)
,  { A } }
Colors of variables: wff set class
Syntax hints:    \/ wo 358    = wceq 1652   {cab 2422    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   {cpr 3815
This theorem is referenced by:  topsn  17000  concompid  17494  usgra1v  21409  esumsn  24456  cvmlift2lem9  24998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-pw 3801  df-sn 3820  df-pr 3821
  Copyright terms: Public domain W3C validator