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Mirrors > Home > ILE Home > Th. List > brun | GIF version |
Description: The union of two binary relations. (Contributed by NM, 21-Dec-2008.) |
Ref | Expression |
---|---|
brun | ⊢ (A(𝑅 ∪ 𝑆)B ↔ (A𝑅B ∨ A𝑆B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 3078 | . 2 ⊢ (〈A, B〉 ∈ (𝑅 ∪ 𝑆) ↔ (〈A, B〉 ∈ 𝑅 ∨ 〈A, B〉 ∈ 𝑆)) | |
2 | df-br 3756 | . 2 ⊢ (A(𝑅 ∪ 𝑆)B ↔ 〈A, B〉 ∈ (𝑅 ∪ 𝑆)) | |
3 | df-br 3756 | . . 3 ⊢ (A𝑅B ↔ 〈A, B〉 ∈ 𝑅) | |
4 | df-br 3756 | . . 3 ⊢ (A𝑆B ↔ 〈A, B〉 ∈ 𝑆) | |
5 | 3, 4 | orbi12i 680 | . 2 ⊢ ((A𝑅B ∨ A𝑆B) ↔ (〈A, B〉 ∈ 𝑅 ∨ 〈A, B〉 ∈ 𝑆)) |
6 | 1, 2, 5 | 3bitr4i 201 | 1 ⊢ (A(𝑅 ∪ 𝑆)B ↔ (A𝑅B ∨ A𝑆B)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∨ wo 628 ∈ wcel 1390 ∪ cun 2909 〈cop 3370 class class class wbr 3755 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-br 3756 |
This theorem is referenced by: dmun 4485 qfto 4657 poleloe 4667 cnvun 4672 coundi 4765 coundir 4766 brdifun 6069 ltxrlt 6882 ltxr 8465 |
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