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Theorem ssintub 4060
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 4058 . 2  |-  ( A 
C_  |^| { x  e.  B  |  A  C_  x }  <->  A. y  e.  {
x  e.  B  |  A  C_  x } A  C_  y )
2 sseq2 3362 . . . 4  |-  ( x  =  y  ->  ( A  C_  x  <->  A  C_  y
) )
32elrab 3084 . . 3  |-  ( y  e.  { x  e.  B  |  A  C_  x }  <->  ( y  e.  B  /\  A  C_  y ) )
43simprbi 451 . 2  |-  ( y  e.  { x  e.  B  |  A  C_  x }  ->  A  C_  y )
51, 4mprgbir 2768 1  |-  A  C_  |^|
{ x  e.  B  |  A  C_  x }
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   {crab 2701    C_ wss 3312   |^|cint 4042
This theorem is referenced by:  intmin  4062  wuncid  8608  mrcssid  13832  lspssid  16051  lbsextlem3  16222  aspssid  16382  sscls  17110  filufint  17942  spanss2  22837  shsval2i  22879  ococin  22900  chsupsn  22905  sssigagen  24518  igenss  26626  rgspnssid  27307  pclssidN  30593  dochocss  32065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-in 3319  df-ss 3326  df-int 4043
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