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Theorem intid 4231
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 snex 4216 . . 3  |-  { A }  e.  _V
2 eleq2 2344 . . . 4  |-  ( x  =  { A }  ->  ( A  e.  x  <->  A  e.  { A }
) )
3 intid.1 . . . . 5  |-  A  e. 
_V
43snid 3667 . . . 4  |-  A  e. 
{ A }
52, 4intmin3 3890 . . 3  |-  ( { A }  e.  _V  ->  |^| { x  |  A  e.  x }  C_ 
{ A } )
61, 5ax-mp 8 . 2  |-  |^| { x  |  A  e.  x }  C_  { A }
73elintab 3873 . . . 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 1537 . . 3  |-  A  e. 
|^| { x  |  A  e.  x }
10 snssi 3759 . . 3  |-  ( A  e.  |^| { x  |  A  e.  x }  ->  { A }  C_  |^|
{ x  |  A  e.  x } )
119, 10ax-mp 8 . 2  |-  { A }  C_  |^| { x  |  A  e.  x }
126, 11eqssi 3195 1  |-  |^| { x  |  A  e.  x }  =  { A }
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    C_ wss 3152   {csn 3640   |^|cint 3862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-int 3863
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