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Definition df-inf 6840
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 6838 . 2 class inf(𝐴, 𝐵, 𝑅)
53ccnv 4508 . . 3 class 𝑅
61, 2, 5csup 6837 . 2 class sup(𝐴, 𝐵, 𝑅)
74, 6wceq 1316 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff set class
This definition is referenced by:  infeq1  6866  infeq2  6869  infeq3  6870  infeq123d  6871  nfinf  6872  eqinfti  6875  infvalti  6877  infclti  6878  inflbti  6879  infglbti  6880  infsnti  6885  inf00  6886  infisoti  6887  dfinfre  8682  infrenegsupex  9357  infxrnegsupex  11000
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