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Definition df-inf 8390
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 8388 . 2 class inf(𝐴, 𝐵, 𝑅)
53ccnv 5142 . . 3 class 𝑅
61, 2, 5csup 8387 . 2 class sup(𝐴, 𝐵, 𝑅)
74, 6wceq 1523 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff setvar class
This definition is referenced by:  infeq1  8423  infeq2  8426  infeq3  8427  infeq123d  8428  nfinf  8429  infexd  8430  eqinf  8431  infval  8433  infcl  8435  inflb  8436  infglb  8437  infglbb  8438  fiinfcl  8448  infltoreq  8449  inf00  8452  infempty  8453  infiso  8454  dfinfre  11042  infrenegsup  11044  tosglb  29798  rencldnfilem  37701
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