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

Theorem snelpw 3977
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
Hypothesis
Ref Expression
snelpw.1  |-  A  e. 
_V
Assertion
Ref Expression
snelpw  |-  ( A  e.  B  <->  { A }  e.  ~P B
)

Proof of Theorem snelpw
StepHypRef Expression
1 snelpw.1 . . 3  |-  A  e. 
_V
21snss 3522 . 2  |-  ( A  e.  B  <->  { A }  C_  B )
3 snexgOLD 3963 . . . 4  |-  ( A  e.  _V  ->  { A }  e.  _V )
41, 3ax-mp 7 . . 3  |-  { A }  e.  _V
54elpw 3393 . 2  |-  ( { A }  e.  ~P B 
<->  { A }  C_  B )
62, 5bitr4i 180 1  |-  ( A  e.  B  <->  { A }  e.  ~P B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 102    e. wcel 1409   _Vcvv 2574    C_ wss 2945   ~Pcpw 3387   {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator