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Theorem pwsnss 3642
Description: The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)
Assertion
Ref Expression
pwsnss  |-  { (/) ,  { A } }  C_ 
~P { A }

Proof of Theorem pwsnss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sssnr 3592 . . 3  |-  ( ( x  =  (/)  \/  x  =  { A } )  ->  x  C_  { A } )
21ss2abi 3091 . 2  |-  { x  |  ( x  =  (/)  \/  x  =  { A } ) }  C_  { x  |  x  C_  { A } }
3 dfpr2 3460 . 2  |-  { (/) ,  { A } }  =  { x  |  ( x  =  (/)  \/  x  =  { A } ) }
4 df-pw 3427 . 2  |-  ~P { A }  =  {
x  |  x  C_  { A } }
52, 3, 43sstr4i 3063 1  |-  { (/) ,  { A } }  C_ 
~P { A }
Colors of variables: wff set class
Syntax hints:    \/ wo 664    = wceq 1289   {cab 2074    C_ wss 2997   (/)c0 3284   ~Pcpw 3425   {csn 3441   {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448
This theorem is referenced by:  pwpw0ss  3643
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