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Theorem intmin 4308
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 3112 . . . . 5  |-  y  e. 
_V
21elintrab 4300 . . . 4  |-  ( y  e.  |^| { x  e.  B  |  A  C_  x }  <->  A. x  e.  B  ( A  C_  x  -> 
y  e.  x ) )
3 ssid 3518 . . . . 5  |-  A  C_  A
4 sseq2 3521 . . . . . . 7  |-  ( x  =  A  ->  ( A  C_  x  <->  A  C_  A
) )
5 eleq2 2530 . . . . . . 7  |-  ( x  =  A  ->  (
y  e.  x  <->  y  e.  A ) )
64, 5imbi12d 320 . . . . . 6  |-  ( x  =  A  ->  (
( A  C_  x  ->  y  e.  x )  <-> 
( A  C_  A  ->  y  e.  A ) ) )
76rspcv 3206 . . . . 5  |-  ( A  e.  B  ->  ( A. x  e.  B  ( A  C_  x  -> 
y  e.  x )  ->  ( A  C_  A  ->  y  e.  A
) ) )
83, 7mpii 43 . . . 4  |-  ( A  e.  B  ->  ( A. x  e.  B  ( A  C_  x  -> 
y  e.  x )  ->  y  e.  A
) )
92, 8syl5bi 217 . . 3  |-  ( A  e.  B  ->  (
y  e.  |^| { x  e.  B  |  A  C_  x }  ->  y  e.  A ) )
109ssrdv 3505 . 2  |-  ( A  e.  B  ->  |^| { x  e.  B  |  A  C_  x }  C_  A
)
11 ssintub 4306 . . 3  |-  A  C_  |^|
{ x  e.  B  |  A  C_  x }
1211a1i 11 . 2  |-  ( A  e.  B  ->  A  C_ 
|^| { x  e.  B  |  A  C_  x }
)
1310, 12eqssd 3516 1  |-  ( A  e.  B  ->  |^| { x  e.  B  |  A  C_  x }  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811    C_ wss 3471   |^|cint 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-in 3478  df-ss 3485  df-int 4289
This theorem is referenced by:  intmin2  4316  ordintdif  4936  bm2.5ii  6640  onsucmin  6655  rankonidlem  8263  rankval4  8302  mrcid  15029  lspid  17754  aspid  18105  cldcls  19669  spanid  26391  chsupid  26456  igenidl2  30624  pclidN  35721  diaocN  36953
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