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Theorem snelpwi 3976
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
Assertion
Ref Expression
snelpwi  |-  ( A  e.  B  ->  { A }  e.  ~P B
)

Proof of Theorem snelpwi
StepHypRef Expression
1 snssi 3536 . 2  |-  ( A  e.  B  ->  { A }  C_  B )
2 elex 2583 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
3 snexgOLD 3963 . . 3  |-  ( A  e.  _V  ->  { A }  e.  _V )
4 elpwg 3395 . . 3  |-  ( { A }  e.  _V  ->  ( { A }  e.  ~P B  <->  { A }  C_  B ) )
52, 3, 43syl 17 . 2  |-  ( A  e.  B  ->  ( { A }  e.  ~P B 
<->  { A }  C_  B ) )
61, 5mpbird 160 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    e. wcel 1409   _Vcvv 2574    C_ wss 2945   ~Pcpw 3387   {csn 3403
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-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-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955
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-in 2952  df-ss 2959  df-pw 3389  df-sn 3409
This theorem is referenced by:  unipw  3981
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