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Theorem supmaxti 6981
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 527 . . 3  |-  ( (
ph  /\  ( y  e.  A  /\  y R C ) )  -> 
y R C )
6 breq2 3993 . . . 4  |-  ( x  =  C  ->  (
y R x  <->  y R C ) )
76rspcev 2834 . . 3  |-  ( ( C  e.  B  /\  y R C )  ->  E. x  e.  B  y R x )
84, 5, 7syl2an2r 590 . 2  |-  ( (
ph  /\  ( y  e.  A  /\  y R C ) )  ->  E. x  e.  B  y R x )
91, 2, 3, 8eqsuptid 6974 1  |-  ( ph  ->  sup ( B ,  A ,  R )  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   E.wrex 2449   class class class wbr 3989   supcsup 6959
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-riota 5809  df-sup 6961
This theorem is referenced by:  supsnti  6982  sup3exmid  8873  maxleim  11169  xrmaxleim  11207  supfz  14100
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