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Theorem snelpwi 4369
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 3902 . 2  |-  ( A  e.  B  ->  { A }  C_  B )
2 snex 4365 . . 3  |-  { A }  e.  _V
32elpw 3765 . 2  |-  ( { A }  e.  ~P B 
<->  { A }  C_  B )
41, 3sylibr 204 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721    C_ wss 3280   ~Pcpw 3759   {csn 3774
This theorem is referenced by:  unipw  4374  canth2  7219  unifpw  7367  marypha1lem  7396  infpwfidom  7865  ackbij1lem4  8059  acsfn  13839  sylow2a  15208  txdis  17617  txdis1cn  17620  symgtgp  18084  esumcst  24408  cntnevol  24535  coinflippvt  24695  onsucsuccmpi  26097  locfindis  26275  lpirlnr  27189  unipwrVD  28653  unipwr  28654  pclfinN  30382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-pw 3761  df-sn 3780  df-pr 3781
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