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Definition df-inf 8209
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 8207 . 2 class inf(𝐴, 𝐵, 𝑅)
53ccnv 5027 . . 3 class 𝑅
61, 2, 5csup 8206 . 2 class sup(𝐴, 𝐵, 𝑅)
74, 6wceq 1474 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff setvar class
This definition is referenced by:  infeq1  8242  infeq2  8245  infeq3  8246  infeq123d  8247  nfinf  8248  infexd  8249  eqinf  8250  infval  8252  infcl  8254  inflb  8255  infglb  8256  infglbb  8257  fiinfcl  8267  infltoreq  8268  inf00  8271  infempty  8272  infiso  8273  lbinf  10825  dfinfre  10851  infrenegsup  10853  tosglb  28807  rencldnfilem  36198
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