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Theorem intmin2 3889
Description: Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
Hypothesis
Ref Expression
intmin2.1  |-  A  e. 
_V
Assertion
Ref Expression
intmin2  |-  |^| { x  |  A  C_  x }  =  A
Distinct variable group:    x, A

Proof of Theorem intmin2
StepHypRef Expression
1 rabab 2805 . . 3  |-  { x  e.  _V  |  A  C_  x }  =  {
x  |  A  C_  x }
21inteqi 3866 . 2  |-  |^| { x  e.  _V  |  A  C_  x }  =  |^| { x  |  A  C_  x }
3 intmin2.1 . . 3  |-  A  e. 
_V
4 intmin 3882 . . 3  |-  ( A  e.  _V  ->  |^| { x  e.  _V  |  A  C_  x }  =  A
)
53, 4ax-mp 8 . 2  |-  |^| { x  e.  _V  |  A  C_  x }  =  A
62, 5eqtr3i 2305 1  |-  |^| { x  |  A  C_  x }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547   _Vcvv 2788    C_ wss 3152   |^|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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-ral 2548  df-rab 2552  df-v 2790  df-in 3159  df-ss 3166  df-int 3863
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