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Theorem supmaxti 6678
Description: The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.)
Hypotheses
Ref Expression
supmaxti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
supmaxti.2  |-  ( ph  ->  C  e.  A )
supmaxti.3  |-  ( ph  ->  C  e.  B )
supmaxti.4  |-  ( (
ph  /\  y  e.  B )  ->  -.  C R y )
Assertion
Ref Expression
supmaxti  |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
Distinct variable groups:    u, A, v, y    u, B, v, y    u, C, v, y    u, R, v, y    ph, u, v, y

Proof of Theorem supmaxti
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 supmaxti.ti . 2  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
2 supmaxti.2 . 2  |-  ( ph  ->  C  e.  A )
3 supmaxti.4 . 2  |-  ( (
ph  /\  y  e.  B )  ->  -.  C R y )
4 supmaxti.3 . . 3  |-  ( ph  ->  C  e.  B )
5 simprr 499 . . 3  |-  ( (
ph  /\  ( y  e.  A  /\  y R C ) )  -> 
y R C )
6 breq2 3841 . . . 4  |-  ( x  =  C  ->  (
y R x  <->  y R C ) )
76rspcev 2722 . . 3  |-  ( ( C  e.  B  /\  y R C )  ->  E. x  e.  B  y R x )
84, 5, 7syl2an2r 562 . 2  |-  ( (
ph  /\  ( y  e.  A  /\  y R C ) )  ->  E. x  e.  B  y R x )
91, 2, 3, 8eqsuptid 6671 1  |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   E.wrex 2360   class class class wbr 3837   supcsup 6656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-riota 5590  df-sup 6658
This theorem is referenced by:  supsnti  6679  maxleim  10603  supfz  11562
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