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Theorem intid 4242
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 4203 . . 3  |-  { A }  e.  _V
3 eleq2 2253 . . . 4  |-  ( x  =  { A }  ->  ( A  e.  x  <->  A  e.  { A }
) )
41snid 3638 . . . 4  |-  A  e. 
{ A }
53, 4intmin3 3886 . . 3  |-  ( { A }  e.  _V  ->  |^| { x  |  A  e.  x }  C_ 
{ A } )
62, 5ax-mp 5 . 2  |-  |^| { x  |  A  e.  x }  C_  { A }
71elintab 3870 . . . 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 1464 . . 3  |-  A  e. 
|^| { x  |  A  e.  x }
10 snssi 3751 . . 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 3186 1  |-  |^| { x  |  A  e.  x }  =  { A }
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160   {cab 2175   _Vcvv 2752    C_ wss 3144   {csn 3607   |^|cint 3859
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  df-int 3860
This theorem is referenced by: (None)
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