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Theorem snelpwi 4220
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 3759 . 2  |-  ( A  e.  B  ->  { A }  C_  B )
2 snex 4216 . . 3  |-  { A }  e.  _V
32elpw 3631 . 2  |-  ( { A }  e.  ~P B 
<->  { A }  C_  B )
41, 3sylibr 203 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   {csn 3640
This theorem is referenced by:  unipw  4224  canth2  7014  unifpw  7158  marypha1lem  7186  infpwfidom  7655  ackbij1lem4  7849  acsfn  13561  sylow2a  14930  txdis  17326  txdis1cn  17329  symgtgp  17784  cntnevol  23175  esumcst  23436  coinflippvt  23685  onsucsuccmpi  24882  iscst4  25177  nZdef  25180  prsubrtr  25399  locfindis  26305  lpirlnr  27321  snelpwrOLD  28607  unipwrVD  28608  unipwr  28609  pclfinN  30089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647
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