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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 4153 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∪ 𝑆) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∨ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
| 2 | df-br 5144 | . 2 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ∪ 𝑆)) | |
| 3 | df-br 5144 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
| 4 | df-br 5144 | . . 3 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
| 5 | 3, 4 | orbi12i 915 | . 2 ⊢ ((𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∨ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
| 6 | 1, 2, 5 | 3bitr4i 303 | 1 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∨ wo 848 ∈ wcel 2108 ∪ cun 3949 〈cop 4632 class class class wbr 5143 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-br 5144 |
| This theorem is referenced by: dmun 5921 qfto 6141 poleloe 6151 cnvun 6162 coundi 6267 coundir 6268 fununmo 6613 eqfunresadj 7380 brdifun 8775 fpwwe2lem12 10682 ltxrlt 11331 ltxr 13157 dfle2 13189 brprop 32706 satfbrsuc 35371 dfso2 35755 dfon3 35893 brcup 35940 dfrdg4 35952 dffrege99 43975 |
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