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Theorem intid 4018
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 3987 . . 3  |-  { A }  e.  _V
3 eleq2 2148 . . . 4  |-  ( x  =  { A }  ->  ( A  e.  x  <->  A  e.  { A }
) )
41snid 3452 . . . 4  |-  A  e. 
{ A }
53, 4intmin3 3692 . . 3  |-  ( { A }  e.  _V  ->  |^| { x  |  A  e.  x }  C_ 
{ A } )
62, 5ax-mp 7 . 2  |-  |^| { x  |  A  e.  x }  C_  { A }
71elintab 3676 . . . 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 1385 . . 3  |-  A  e. 
|^| { x  |  A  e.  x }
10 snssi 3558 . . 3  |-  ( A  e.  |^| { x  |  A  e.  x }  ->  { A }  C_  |^|
{ x  |  A  e.  x } )
119, 10ax-mp 7 . 2  |-  { A }  C_  |^| { x  |  A  e.  x }
126, 11eqssi 3028 1  |-  |^| { x  |  A  e.  x }  =  { A }
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436   {cab 2071   _Vcvv 2614    C_ wss 2986   {csn 3425   |^|cint 3665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-int 3666
This theorem is referenced by: (None)
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