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Theorem intid 4630
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 4613 . . 3  |-  { A }  e.  _V
2 eleq2 2518 . . . 4  |-  ( x  =  { A }  ->  ( A  e.  x  <->  A  e.  { A }
) )
3 intid.1 . . . . 5  |-  A  e. 
_V
43snid 3963 . . . 4  |-  A  e. 
{ A }
52, 4intmin3 4232 . . 3  |-  ( { A }  e.  _V  ->  |^| { x  |  A  e.  x }  C_ 
{ A } )
61, 5ax-mp 5 . 2  |-  |^| { x  |  A  e.  x }  C_  { A }
73elintab 4214 . . . 4  |-  ( A  e.  |^| { x  |  A  e.  x }  <->  A. x ( A  e.  x  ->  A  e.  x ) )
8 id 22 . . . 4  |-  ( A  e.  x  ->  A  e.  x )
97, 8mpgbir 1676 . . 3  |-  A  e. 
|^| { x  |  A  e.  x }
10 snssi 4084 . . 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 3415 1  |-  |^| { x  |  A  e.  x }  =  { A }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1447    e. wcel 1890   {cab 2437   _Vcvv 3012    C_ wss 3371   {csn 3935   |^|cint 4203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pr 4611
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-v 3014  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-sn 3936  df-pr 3938  df-int 4204
This theorem is referenced by: (None)
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