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Definition df-inf 6627
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 6625 . 2 class inf(𝐴, 𝐵, 𝑅)
53ccnv 4412 . . 3 class 𝑅
61, 2, 5csup 6624 . 2 class sup(𝐴, 𝐵, 𝑅)
74, 6wceq 1287 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff set class
This definition is referenced by:  infeq1  6653  infeq2  6656  infeq3  6657  infeq123d  6658  nfinf  6659  eqinfti  6662  infvalti  6664  infclti  6665  inflbti  6666  infglbti  6667  infsnti  6672  inf00  6673  infisoti  6674  dfinfre  8355  infrenegsupex  9017
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