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Theorem ssintub 3633
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 3631 . 2  |-  ( A 
C_  |^| { x  e.  B  |  A  C_  x }  <->  A. y  e.  {
x  e.  B  |  A  C_  x } A  C_  y )
2 sseq2 2967 . . . 4  |-  ( x  =  y  ->  ( A  C_  x  <->  A  C_  y
) )
32elrab 2698 . . 3  |-  ( y  e.  { x  e.  B  |  A  C_  x }  <->  ( y  e.  B  /\  A  C_  y ) )
43simprbi 260 . 2  |-  ( y  e.  { x  e.  B  |  A  C_  x }  ->  A  C_  y )
51, 4mprgbir 2379 1  |-  A  C_  |^|
{ x  e.  B  |  A  C_  x }
Colors of variables: wff set class
Syntax hints:    e. wcel 1393   {crab 2310    C_ wss 2917   |^|cint 3615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-int 3616
This theorem is referenced by:  intmin  3635
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