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

Theorem snelpwi 4129
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 3659 . 2  |-  ( A  e.  B  ->  { A }  C_  B )
2 elex 2692 . . 3  |-  ( A  e.  B  ->  A  e.  _V )
3 snexg 4103 . . 3  |-  ( A  e.  _V  ->  { A }  e.  _V )
4 elpwg 3513 . . 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 166 1  |-  ( A  e.  B  ->  { A }  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1480   _Vcvv 2681    C_ wss 3066   ~Pcpw 3505   {csn 3522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528
This theorem is referenced by:  unipw  4134  infpwfidom  7047  txdis  12435  txdis1cn  12436
  Copyright terms: Public domain W3C validator