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Definition df-inf 7957
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 7955 . 2  class inf ( A ,  B ,  R
)
53ccnv 4833 . . 3  class  `' R
61, 2, 5csup 7954 . 2  class  sup ( A ,  B ,  `' R )
74, 6wceq 1444 1  wff inf ( A ,  B ,  R
)  =  sup ( A ,  B ,  `' R )
Colors of variables: wff setvar class
This definition is referenced by:  infeq1  7992  infeq2  7995  infeq3  7996  infeq123d  7997  nfinf  7998  infexd  7999  eqinf  8000  infval  8002  infcl  8004  inflb  8005  infglb  8006  infglbb  8007  fiinfcl  8017  infltoreq  8018  inf00  8021  infempty  8022  infiso  8023  lbinf  10559  dfinfre  10588  infrenegsup  10591  tosglb  28431  rencldnfilem  35663
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