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Theorem supmaxti 6840
 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
supmaxti.2
supmaxti.3
supmaxti.4
Assertion
Ref Expression
supmaxti
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,

Proof of Theorem supmaxti
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 supmaxti.ti . 2
2 supmaxti.2 . 2
3 supmaxti.4 . 2
4 supmaxti.3 . . 3
5 simprr 504 . . 3
6 breq2 3897 . . . 4
76rspcev 2758 . . 3
84, 5, 7syl2an2r 567 . 2
91, 2, 3, 8eqsuptid 6833 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104   wceq 1312   wcel 1461  wrex 2389   class class class wbr 3893  csup 6818 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095 This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-un 3039  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-iota 5044  df-riota 5682  df-sup 6820 This theorem is referenced by:  supsnti  6841  sup3exmid  8618  maxleim  10862  xrmaxleim  10898  supfz  12914
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