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Definition df-inf 6889
 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 6887 . 2 class inf(𝐴, 𝐵, 𝑅)
53ccnv 4550 . . 3 class 𝑅
61, 2, 5csup 6886 . 2 class sup(𝐴, 𝐵, 𝑅)
74, 6wceq 1332 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
 Colors of variables: wff set class This definition is referenced by:  infeq1  6915  infeq2  6918  infeq3  6919  infeq123d  6920  nfinf  6921  eqinfti  6924  infvalti  6926  infclti  6927  inflbti  6928  infglbti  6929  infsnti  6934  inf00  6935  infisoti  6936  dfinfre  8767  infrenegsupex  9445  infxrnegsupex  11093
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