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Theorem snelpwi 4536
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 4016 . 2  |-  ( A  e.  B  ->  { A }  C_  B )
2 snex 4532 . . 3  |-  { A }  e.  _V
32elpw 3865 . 2  |-  ( { A }  e.  ~P B 
<->  { A }  C_  B )
41, 3sylibr 212 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756    C_ wss 3327   ~Pcpw 3859   {csn 3876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-pw 3861  df-sn 3877  df-pr 3879
This theorem is referenced by:  unipw  4541  canth2  7463  unifpw  7613  marypha1lem  7682  infpwfidom  8197  ackbij1lem4  8391  acsfn  14596  sylow2a  16117  txdis  19204  txdis1cn  19207  symgtgp  19671  esumcst  26513  cntnevol  26641  coinflippvt  26866  onsucsuccmpi  28288  locfindis  28575  lpirlnr  29471  lincvalsng  30948  snlindsntor  31003  unipwrVD  31566  unipwr  31567  pclfinN  33542
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