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Mirrors > Home > MPE Home > Th. List > df-inf | Structured version Visualization version 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 8894 | . 2 class inf(𝐴, 𝐵, 𝑅) |
5 | 3 | ccnv 5548 | . . 3 class ◡𝑅 |
6 | 1, 2, 5 | csup 8893 | . 2 class sup(𝐴, 𝐵, ◡𝑅) |
7 | 4, 6 | wceq 1528 | 1 wff inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) |
Colors of variables: wff setvar class |
This definition is referenced by: infeq1 8929 infeq2 8932 infeq3 8933 infeq123d 8934 nfinf 8935 infexd 8936 eqinf 8937 infval 8939 infcl 8941 inflb 8942 infglb 8943 infglbb 8944 fiinfcl 8954 infltoreq 8955 inf00 8959 infempty 8960 infiso 8961 dfinfre 11611 infrenegsup 11613 tosglb 30585 rencldnfilem 39297 |
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