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Theorem intmin 3759
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 2661 . . . . 5  |-  y  e. 
_V
21elintrab 3751 . . . 4  |-  ( y  e.  |^| { x  e.  B  |  A  C_  x }  <->  A. x  e.  B  ( A  C_  x  -> 
y  e.  x ) )
3 ssid 3085 . . . . 5  |-  A  C_  A
4 sseq2 3089 . . . . . . 7  |-  ( x  =  A  ->  ( A  C_  x  <->  A  C_  A
) )
5 eleq2 2179 . . . . . . 7  |-  ( x  =  A  ->  (
y  e.  x  <->  y  e.  A ) )
64, 5imbi12d 233 . . . . . 6  |-  ( x  =  A  ->  (
( A  C_  x  ->  y  e.  x )  <-> 
( A  C_  A  ->  y  e.  A ) ) )
76rspcv 2757 . . . . 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, 8syl5bi 151 . . 3  |-  ( A  e.  B  ->  (
y  e.  |^| { x  e.  B  |  A  C_  x }  ->  y  e.  A ) )
109ssrdv 3071 . 2  |-  ( A  e.  B  ->  |^| { x  e.  B  |  A  C_  x }  C_  A
)
11 ssintub 3757 . . 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 3082 1  |-  ( A  e.  B  ->  |^| { x  e.  B  |  A  C_  x }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463   A.wral 2391   {crab 2395    C_ wss 3039   |^|cint 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rab 2400  df-v 2660  df-in 3045  df-ss 3052  df-int 3740
This theorem is referenced by:  intmin2  3765  bm2.5ii  4380  onsucmin  4391  cldcls  12178
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