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Theorem supmaxti 6950
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 522 . . 3  |-  ( (
ph  /\  ( y  e.  A  /\  y R C ) )  -> 
y R C )
6 breq2 3971 . . . 4  |-  ( x  =  C  ->  (
y R x  <->  y R C ) )
76rspcev 2816 . . 3  |-  ( ( C  e.  B  /\  y R C )  ->  E. x  e.  B  y R x )
84, 5, 7syl2an2r 585 . 2  |-  ( (
ph  /\  ( y  e.  A  /\  y R C ) )  ->  E. x  e.  B  y R x )
91, 2, 3, 8eqsuptid 6943 1  |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   E.wrex 2436   class class class wbr 3967   supcsup 6928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-un 3106  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-iota 5137  df-riota 5782  df-sup 6930
This theorem is referenced by:  supsnti  6951  sup3exmid  8833  maxleim  11116  xrmaxleim  11152  supfz  13710
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