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Theorem intmin 3864
Description: Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
intmin  |-  ( A  e.  B  ->  |^| { x  e.  B  |  A  C_  x }  =  A )
Distinct variable groups:    x, A    x, B

Proof of Theorem intmin
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2740 . . . . 5  |-  y  e. 
_V
21elintrab 3856 . . . 4  |-  ( y  e.  |^| { x  e.  B  |  A  C_  x }  <->  A. x  e.  B  ( A  C_  x  -> 
y  e.  x ) )
3 ssid 3175 . . . . 5  |-  A  C_  A
4 sseq2 3179 . . . . . . 7  |-  ( x  =  A  ->  ( A  C_  x  <->  A  C_  A
) )
5 eleq2 2241 . . . . . . 7  |-  ( x  =  A  ->  (
y  e.  x  <->  y  e.  A ) )
64, 5imbi12d 234 . . . . . 6  |-  ( x  =  A  ->  (
( A  C_  x  ->  y  e.  x )  <-> 
( A  C_  A  ->  y  e.  A ) ) )
76rspcv 2837 . . . . 5  |-  ( A  e.  B  ->  ( A. x  e.  B  ( A  C_  x  -> 
y  e.  x )  ->  ( A  C_  A  ->  y  e.  A
) ) )
83, 7mpii 44 . . . 4  |-  ( A  e.  B  ->  ( A. x  e.  B  ( A  C_  x  -> 
y  e.  x )  ->  y  e.  A
) )
92, 8biimtrid 152 . . 3  |-  ( A  e.  B  ->  (
y  e.  |^| { x  e.  B  |  A  C_  x }  ->  y  e.  A ) )
109ssrdv 3161 . 2  |-  ( A  e.  B  ->  |^| { x  e.  B  |  A  C_  x }  C_  A
)
11 ssintub 3862 . . 3  |-  A  C_  |^|
{ x  e.  B  |  A  C_  x }
1211a1i 9 . 2  |-  ( A  e.  B  ->  A  C_ 
|^| { x  e.  B  |  A  C_  x }
)
1310, 12eqssd 3172 1  |-  ( A  e.  B  ->  |^| { x  e.  B  |  A  C_  x }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459    C_ wss 3129   |^|cint 3844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2739  df-in 3135  df-ss 3142  df-int 3845
This theorem is referenced by:  intmin2  3870  bm2.5ii  4495  onsucmin  4506  cldcls  13584
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