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Definition df-inf 7975
Description: Define the infimum of class  A. It is meaningful when  R is a relation that strictly orders 
B and when the infimum exists. For example,  R could be 'less than',  B could be the set of real numbers, and  A could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
Assertion
Ref Expression
df-inf  |- inf ( A ,  B ,  R
)  =  sup ( A ,  B ,  `' R )

Detailed syntax breakdown of Definition df-inf
StepHypRef Expression
1 cA . . 3  class  A
2 cB . . 3  class  B
3 cR . . 3  class  R
41, 2, 3cinf 7973 . 2  class inf ( A ,  B ,  R
)
53ccnv 4838 . . 3  class  `' R
61, 2, 5csup 7972 . 2  class  sup ( A ,  B ,  `' R )
74, 6wceq 1452 1  wff inf ( A ,  B ,  R
)  =  sup ( A ,  B ,  `' R )
Colors of variables: wff setvar class
This definition is referenced by:  infeq1  8010  infeq2  8013  infeq3  8014  infeq123d  8015  nfinf  8016  infexd  8017  eqinf  8018  infval  8020  infcl  8022  inflb  8023  infglb  8024  infglbb  8025  fiinfcl  8035  infltoreq  8036  inf00  8039  infempty  8040  infiso  8041  lbinf  10581  dfinfre  10610  infrenegsup  10613  tosglb  28506  rencldnfilem  35734
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