ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  snelpwi Unicode version

Theorem snelpwi 4230
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 3751 . 2  |-  ( A  e.  B  ->  { A }  C_  B )
2 elex 2763 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
3 snexg 4202 . . 3  |-  ( A  e.  _V  ->  { A }  e.  _V )
4 elpwg 3598 . . 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 167 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    e. wcel 2160   _Vcvv 2752    C_ wss 3144   ~Pcpw 3590   {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613
This theorem is referenced by:  unipw  4235  infpwfidom  7226  txdis  14229  txdis1cn  14230
  Copyright terms: Public domain W3C validator