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Theorem supmaxti 7002
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 531 . . 3  |-  ( (
ph  /\  ( y  e.  A  /\  y R C ) )  -> 
y R C )
6 breq2 4007 . . . 4  |-  ( x  =  C  ->  (
y R x  <->  y R C ) )
76rspcev 2841 . . 3  |-  ( ( C  e.  B  /\  y R C )  ->  E. x  e.  B  y R x )
84, 5, 7syl2an2r 595 . 2  |-  ( (
ph  /\  ( y  e.  A  /\  y R C ) )  ->  E. x  e.  B  y R x )
91, 2, 3, 8eqsuptid 6995 1  |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   E.wrex 2456   class class class wbr 4003   supcsup 6980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-riota 5830  df-sup 6982
This theorem is referenced by:  supsnti  7003  sup3exmid  8912  maxleim  11209  xrmaxleim  11247  supfz  14700
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