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Description: Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of TakeutiZaring p. 29 generalized to arbitrary classes. Class R often denotes a relation such as " < " that compares two classes A and B , which might be numbers such as 1 and 2 (see df-ltxr for the specific definition of < ). As a wff, relations are true or false. For example, ( R = { <. 2 , 6 >. , <. 3 , 9 >. } -> 3 R 9 ) ( ex-br ). Often class R meets the Rel criteria to be defined in df-rel , and in particular R may be a function (see df-fun ). This definition of relations is well-defined, although not very meaningful, when classes A and/or B are proper classes (i.e., are not sets). On the other hand, we often find uses for this definition when R is a proper class (see for example iprc ). (Contributed by NM, 31-Dec-1993)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-br | ⊢ ( 𝐴 𝑅 𝐵 ↔ 〈 𝐴 , 𝐵 〉 ∈ 𝑅 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | cA | ⊢ 𝐴 | |
| 1 | cR | ⊢ 𝑅 | |
| 2 | cB | ⊢ 𝐵 | |
| 3 | 0 2 1 | wbr | ⊢ 𝐴 𝑅 𝐵 |
| 4 | 0 2 | cop | ⊢ 〈 𝐴 , 𝐵 〉 |
| 5 | 4 1 | wcel | ⊢ 〈 𝐴 , 𝐵 〉 ∈ 𝑅 |
| 6 | 3 5 | wb | ⊢ ( 𝐴 𝑅 𝐵 ↔ 〈 𝐴 , 𝐵 〉 ∈ 𝑅 ) |