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Theorem intmin2 4077
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 2973 . . 3  |-  { x  e.  _V  |  A  C_  x }  =  {
x  |  A  C_  x }
21inteqi 4054 . 2  |-  |^| { x  e.  _V  |  A  C_  x }  =  |^| { x  |  A  C_  x }
3 intmin2.1 . . 3  |-  A  e. 
_V
4 intmin 4070 . . 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 2458 1  |-  |^| { x  |  A  C_  x }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   {cab 2422   {crab 2709   _Vcvv 2956    C_ wss 3320   |^|cint 4050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rab 2714  df-v 2958  df-in 3327  df-ss 3334  df-int 4051
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