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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 4124 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∪ 𝑆) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∨ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
2 | df-br 5059 | . 2 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ∪ 𝑆)) | |
3 | df-br 5059 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
4 | df-br 5059 | . . 3 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
5 | 3, 4 | orbi12i 911 | . 2 ⊢ ((𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∨ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
6 | 1, 2, 5 | 3bitr4i 305 | 1 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∨ wo 843 ∈ wcel 2110 ∪ cun 3933 〈cop 4566 class class class wbr 5058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3940 df-br 5059 |
This theorem is referenced by: dmun 5773 qfto 5975 poleloe 5985 cnvun 5995 coundi 6094 coundir 6095 fununmo 6395 brdifun 8312 fpwwe2lem13 10058 ltxrlt 10705 ltxr 12504 dfle2 12534 brprop 30427 satfbrsuc 32608 dfso2 32985 eqfunresadj 32999 dfon3 33348 brcup 33395 dfrdg4 33407 dffrege99 40301 |
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