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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 3084 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∪ 𝑆) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∨ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
2 | df-br 3765 | . 2 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ∪ 𝑆)) | |
3 | df-br 3765 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
4 | df-br 3765 | . . 3 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
5 | 3, 4 | orbi12i 681 | . 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 629 ∈ wcel 1393 ∪ cun 2915 〈cop 3378 class class class wbr 3764 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-br 3765 |
This theorem is referenced by: dmun 4542 qfto 4714 poleloe 4724 cnvun 4729 coundi 4822 coundir 4823 brdifun 6133 ltxrlt 7085 ltxr 8695 |
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