MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snelpwi Structured version   Unicode version

Theorem snelpwi 4409
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 3942 . 2  |-  ( A  e.  B  ->  { A }  C_  B )
2 snex 4405 . . 3  |-  { A }  e.  _V
32elpw 3805 . 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 1725    C_ wss 3320   ~Pcpw 3799   {csn 3814
This theorem is referenced by:  unipw  4414  canth2  7260  unifpw  7409  marypha1lem  7438  infpwfidom  7909  ackbij1lem4  8103  acsfn  13884  sylow2a  15253  txdis  17664  txdis1cn  17667  symgtgp  18131  esumcst  24455  cntnevol  24582  coinflippvt  24742  onsucsuccmpi  26193  locfindis  26385  lpirlnr  27298  unipwrVD  28944  unipwr  28945  pclfinN  30697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-pw 3801  df-sn 3820  df-pr 3821
  Copyright terms: Public domain W3C validator