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Theorem intid 4247
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 4232 . . 3  |-  { A }  e.  _V
2 eleq2 2357 . . . 4  |-  ( x  =  { A }  ->  ( A  e.  x  <->  A  e.  { A }
) )
3 intid.1 . . . . 5  |-  A  e. 
_V
43snid 3680 . . . 4  |-  A  e. 
{ A }
52, 4intmin3 3906 . . 3  |-  ( { A }  e.  _V  ->  |^| { x  |  A  e.  x }  C_ 
{ A } )
61, 5ax-mp 8 . 2  |-  |^| { x  |  A  e.  x }  C_  { A }
73elintab 3889 . . . 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 1540 . . 3  |-  A  e. 
|^| { x  |  A  e.  x }
10 snssi 3775 . . 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 3208 1  |-  |^| { x  |  A  e.  x }  =  { A }
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    C_ wss 3165   {csn 3653   |^|cint 3878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660  df-int 3879
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