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Theorem pwsnss 3602
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 3552 . . 3  |-  ( ( x  =  (/)  \/  x  =  { A } )  ->  x  C_  { A } )
21ss2abi 3040 . 2  |-  { x  |  ( x  =  (/)  \/  x  =  { A } ) }  C_  { x  |  x  C_  { A } }
3 dfpr2 3422 . 2  |-  { (/) ,  { A } }  =  { x  |  ( x  =  (/)  \/  x  =  { A } ) }
4 df-pw 3389 . 2  |-  ~P { A }  =  {
x  |  x  C_  { A } }
52, 3, 43sstr4i 3012 1  |-  { (/) ,  { A } }  C_ 
~P { A }
Colors of variables: wff set class
Syntax hints:    \/ wo 639    = wceq 1259   {cab 2042    C_ wss 2945   (/)c0 3252   ~Pcpw 3387   {csn 3403   {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410
This theorem is referenced by:  pwpw0ss  3603
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