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Theorem ssintub 4216
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 4214 . 2  |-  ( A 
C_  |^| { x  e.  B  |  A  C_  x }  <->  A. y  e.  {
x  e.  B  |  A  C_  x } A  C_  y )
2 sseq2 3429 . . . 4  |-  ( x  =  y  ->  ( A  C_  x  <->  A  C_  y
) )
32elrab 3171 . . 3  |-  ( y  e.  { x  e.  B  |  A  C_  x }  <->  ( y  e.  B  /\  A  C_  y ) )
43simprbi 465 . 2  |-  ( y  e.  { x  e.  B  |  A  C_  x }  ->  A  C_  y )
51, 4mprgbir 2729 1  |-  A  C_  |^|
{ x  e.  B  |  A  C_  x }
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1872   {crab 2718    C_ wss 3379   |^|cint 4198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rab 2723  df-v 3024  df-in 3386  df-ss 3393  df-int 4199
This theorem is referenced by:  intmin  4218  wuncid  9119  mrcssid  15466  lspssid  18151  lbsextlem3  18326  aspssid  18500  sscls  20013  filufint  20877  spanss2  26940  shsval2i  26982  ococin  27003  chsupsn  27008  sssigagen  28919  dynkin  28941  igenss  32202  pclssidN  33372  dochocss  34846  rgspnssid  35949
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