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Theorem intmin2 3697
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 2634 . . 3  |-  { x  e.  _V  |  A  C_  x }  =  {
x  |  A  C_  x }
21inteqi 3675 . 2  |-  |^| { x  e.  _V  |  A  C_  x }  =  |^| { x  |  A  C_  x }
3 intmin2.1 . . 3  |-  A  e. 
_V
4 intmin 3691 . . 3  |-  ( A  e.  _V  ->  |^| { x  e.  _V  |  A  C_  x }  =  A
)
53, 4ax-mp 7 . 2  |-  |^| { x  e.  _V  |  A  C_  x }  =  A
62, 5eqtr3i 2107 1  |-  |^| { x  |  A  C_  x }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1287    e. wcel 1436   {cab 2071   {crab 2359   _Vcvv 2615    C_ wss 2988   |^|cint 3671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rab 2364  df-v 2617  df-in 2994  df-ss 3001  df-int 3672
This theorem is referenced by: (None)
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