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Theorem ssintub 3896
Description: Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)
Assertion
Ref Expression
ssintub  |-  A  C_  |^|
{ x  e.  B  |  A  C_  x }
Distinct variable groups:    x, A    x, B

Proof of Theorem ssintub
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssint 3894 . 2  |-  ( A 
C_  |^| { x  e.  B  |  A  C_  x }  <->  A. y  e.  {
x  e.  B  |  A  C_  x } A  C_  y )
2 sseq2 3213 . . . 4  |-  ( x  =  y  ->  ( A  C_  x  <->  A  C_  y
) )
32elrab 2936 . . 3  |-  ( y  e.  { x  e.  B  |  A  C_  x }  <->  ( y  e.  B  /\  A  C_  y ) )
43simprbi 450 . 2  |-  ( y  e.  { x  e.  B  |  A  C_  x }  ->  A  C_  y )
51, 4mprgbir 2626 1  |-  A  C_  |^|
{ x  e.  B  |  A  C_  x }
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   {crab 2560    C_ wss 3165   |^|cint 3878
This theorem is referenced by:  intmin  3898  mrcssid  13535  lspssid  15758  lbsextlem3  15929  aspssid  16089  sscls  16809  filufint  17631  spanss2  21940  shsval2i  21982  ococin  22003  chsupsn  22008  sssigagen  23521  igenss  26790  rgspnssid  27478  pclssidN  30706  dochocss  32178
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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