ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-inf GIF version

Definition df-inf 6977
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 6975 . 2 class inf(𝐴, 𝐵, 𝑅)
53ccnv 4621 . . 3 class 𝑅
61, 2, 5csup 6974 . 2 class sup(𝐴, 𝐵, 𝑅)
74, 6wceq 1353 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff set class
This definition is referenced by:  infeq1  7003  infeq2  7006  infeq3  7007  infeq123d  7008  nfinf  7009  eqinfti  7012  infvalti  7014  infclti  7015  inflbti  7016  infglbti  7017  infsnti  7022  inf00  7023  infisoti  7024  infex2g  7026  dfinfre  8889  infrenegsupex  9570  infxrnegsupex  11242
  Copyright terms: Public domain W3C validator