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Mirrors > Home > MPE Home > Th. List > brun | Structured version Visualization version 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 3715 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∪ 𝑆) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∨ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
2 | df-br 4584 | . 2 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ∪ 𝑆)) | |
3 | df-br 4584 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
4 | df-br 4584 | . . 3 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
5 | 3, 4 | orbi12i 542 | . 2 ⊢ ((𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∨ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
6 | 1, 2, 5 | 3bitr4i 291 | 1 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 ∈ wcel 1977 ∪ cun 3538 〈cop 4131 class class class wbr 4583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-br 4584 |
This theorem is referenced by: dmun 5253 qfto 5436 poleloe 5446 cnvun 5457 coundi 5553 coundir 5554 fununmo 5847 brdifun 7658 fpwwe2lem13 9343 ltxrlt 9987 ltxr 11825 dfle2 11856 dfso2 30897 dfon3 31169 brcup 31216 dfrdg4 31228 dffrege99 37276 |
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