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Mirrors > Home > NFE 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 3220 | . 2 ⊢ (〈A, B〉 ∈ (R ∪ S) ↔ (〈A, B〉 ∈ R ∨ 〈A, B〉 ∈ S)) | |
2 | df-br 4640 | . 2 ⊢ (A(R ∪ S)B ↔ 〈A, B〉 ∈ (R ∪ S)) | |
3 | df-br 4640 | . . 3 ⊢ (ARB ↔ 〈A, B〉 ∈ R) | |
4 | df-br 4640 | . . 3 ⊢ (ASB ↔ 〈A, B〉 ∈ S) | |
5 | 3, 4 | orbi12i 507 | . 2 ⊢ ((ARB ∨ ASB) ↔ (〈A, B〉 ∈ R ∨ 〈A, B〉 ∈ S)) |
6 | 1, 2, 5 | 3bitr4i 268 | 1 ⊢ (A(R ∪ S)B ↔ (ARB ∨ ASB)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∨ wo 357 ∈ wcel 1710 ∪ cun 3207 〈cop 4561 class class class wbr 4639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-br 4640 |
This theorem is referenced by: dmun 4912 cnvun 5033 coundi 5082 coundir 5083 nchoicelem16 6304 |
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