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Definition df-inf 6457
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 6455 . 2 class inf(𝐴, 𝐵, 𝑅)
53ccnv 4370 . . 3 class 𝑅
61, 2, 5csup 6454 . 2 class sup(𝐴, 𝐵, 𝑅)
74, 6wceq 1285 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff set class
This definition is referenced by:  infeq1  6483  infeq2  6486  infeq3  6487  infeq123d  6488  nfinf  6489  eqinfti  6492  infvalti  6494  infclti  6495  inflbti  6496  infglbti  6497  infsnti  6502  inf00  6503  infisoti  6504  dfinfre  8101  infrenegsupex  8763
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