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Theorem intid 4695
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 4678 . . 3  |-  { A }  e.  _V
2 eleq2 2527 . . . 4  |-  ( x  =  { A }  ->  ( A  e.  x  <->  A  e.  { A }
) )
3 intid.1 . . . . 5  |-  A  e. 
_V
43snid 4044 . . . 4  |-  A  e. 
{ A }
52, 4intmin3 4300 . . 3  |-  ( { A }  e.  _V  ->  |^| { x  |  A  e.  x }  C_ 
{ A } )
61, 5ax-mp 5 . 2  |-  |^| { x  |  A  e.  x }  C_  { A }
73elintab 4282 . . . 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 1627 . . 3  |-  A  e. 
|^| { x  |  A  e.  x }
10 snssi 4160 . . 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 3505 1  |-  |^| { x  |  A  e.  x }  =  { A }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   {cab 2439   _Vcvv 3106    C_ wss 3461   {csn 4016   |^|cint 4271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019  df-int 4272
This theorem is referenced by: (None)
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