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Theorem intmin2 3905
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 2818 . . 3  |-  { x  e.  _V  |  A  C_  x }  =  {
x  |  A  C_  x }
21inteqi 3882 . 2  |-  |^| { x  e.  _V  |  A  C_  x }  =  |^| { x  |  A  C_  x }
3 intmin2.1 . . 3  |-  A  e. 
_V
4 intmin 3898 . . 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 2318 1  |-  |^| { x  |  A  C_  x }  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560   _Vcvv 2801    C_ wss 3165   |^|cint 3878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-int 3879
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