ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  intmin Unicode version

Theorem intmin 3828
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 2715 . . . . 5  |-  y  e. 
_V
21elintrab 3820 . . . 4  |-  ( y  e.  |^| { x  e.  B  |  A  C_  x }  <->  A. x  e.  B  ( A  C_  x  -> 
y  e.  x ) )
3 ssid 3148 . . . . 5  |-  A  C_  A
4 sseq2 3152 . . . . . . 7  |-  ( x  =  A  ->  ( A  C_  x  <->  A  C_  A
) )
5 eleq2 2221 . . . . . . 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 2812 . . . . 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 3134 . 2  |-  ( A  e.  B  ->  |^| { x  e.  B  |  A  C_  x }  C_  A
)
11 ssintub 3826 . . 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 3145 1  |-  ( A  e.  B  ->  |^| { x  e.  B  |  A  C_  x }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    e. wcel 2128   A.wral 2435   {crab 2439    C_ wss 3102   |^|cint 3808
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-in 3108  df-ss 3115  df-int 3809
This theorem is referenced by:  intmin2  3834  bm2.5ii  4456  onsucmin  4467  cldcls  12556
  Copyright terms: Public domain W3C validator