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Mirrors > Home > ILE Home > Th. List > df-inf | GIF version |
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.) |
Ref | Expression |
---|---|
df-inf | ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cB | . . 3 class 𝐵 | |
3 | cR | . . 3 class 𝑅 | |
4 | 1, 2, 3 | cinf 6975 | . 2 class inf(𝐴, 𝐵, 𝑅) |
5 | 3 | ccnv 4621 | . . 3 class ◡𝑅 |
6 | 1, 2, 5 | csup 6974 | . 2 class sup(𝐴, 𝐵, ◡𝑅) |
7 | 4, 6 | wceq 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 |
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