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Definition df-inf 8584
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 8582 . 2 class inf(𝐴, 𝐵, 𝑅)
53ccnv 5310 . . 3 class 𝑅
61, 2, 5csup 8581 . 2 class sup(𝐴, 𝐵, 𝑅)
74, 6wceq 1637 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff setvar class
This definition is referenced by:  infeq1  8617  infeq2  8620  infeq3  8621  infeq123d  8622  nfinf  8623  infexd  8624  eqinf  8625  infval  8627  infcl  8629  inflb  8630  infglb  8631  infglbb  8632  fiinfcl  8642  infltoreq  8643  inf00  8646  infempty  8647  infiso  8648  dfinfre  11285  infrenegsup  11287  tosglb  29991  rencldnfilem  37880
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