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Theorem intid 4116
Description: The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
Hypothesis
Ref Expression
intid.1  |-  A  e. 
_V
Assertion
Ref Expression
intid  |-  |^| { x  |  A  e.  x }  =  { A }
Distinct variable group:    x, A

Proof of Theorem intid
StepHypRef Expression
1 intid.1 . . . 4  |-  A  e. 
_V
21snex 4079 . . 3  |-  { A }  e.  _V
3 eleq2 2181 . . . 4  |-  ( x  =  { A }  ->  ( A  e.  x  <->  A  e.  { A }
) )
41snid 3526 . . . 4  |-  A  e. 
{ A }
53, 4intmin3 3768 . . 3  |-  ( { A }  e.  _V  ->  |^| { x  |  A  e.  x }  C_ 
{ A } )
62, 5ax-mp 5 . 2  |-  |^| { x  |  A  e.  x }  C_  { A }
71elintab 3752 . . . 4  |-  ( A  e.  |^| { x  |  A  e.  x }  <->  A. x ( A  e.  x  ->  A  e.  x ) )
8 id 19 . . . 4  |-  ( A  e.  x  ->  A  e.  x )
97, 8mpgbir 1414 . . 3  |-  A  e. 
|^| { x  |  A  e.  x }
10 snssi 3634 . . 3  |-  ( A  e.  |^| { x  |  A  e.  x }  ->  { A }  C_  |^|
{ x  |  A  e.  x } )
119, 10ax-mp 5 . 2  |-  { A }  C_  |^| { x  |  A  e.  x }
126, 11eqssi 3083 1  |-  |^| { x  |  A  e.  x }  =  { A }
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   {cab 2103   _Vcvv 2660    C_ wss 3041   {csn 3497   |^|cint 3741
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-int 3742
This theorem is referenced by: (None)
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