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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 6838 | . 2 class inf(𝐴, 𝐵, 𝑅) |
5 | 3 | ccnv 4508 | . . 3 class ◡𝑅 |
6 | 1, 2, 5 | csup 6837 | . 2 class sup(𝐴, 𝐵, ◡𝑅) |
7 | 4, 6 | wceq 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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