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Theorem brun 3801
 Description: The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
Assertion
Ref Expression
brun (A(𝑅𝑆)B ↔ (A𝑅B A𝑆B))

Proof of Theorem brun
StepHypRef Expression
1 elun 3078 . 2 (⟨A, B (𝑅𝑆) ↔ (⟨A, B 𝑅 A, B 𝑆))
2 df-br 3756 . 2 (A(𝑅𝑆)B ↔ ⟨A, B (𝑅𝑆))
3 df-br 3756 . . 3 (A𝑅B ↔ ⟨A, B 𝑅)
4 df-br 3756 . . 3 (A𝑆B ↔ ⟨A, B 𝑆)
53, 4orbi12i 680 . 2 ((A𝑅B A𝑆B) ↔ (⟨A, B 𝑅 A, B 𝑆))
61, 2, 53bitr4i 201 1 (A(𝑅𝑆)B ↔ (A𝑅B A𝑆B))
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∨ wo 628   ∈ wcel 1390   ∪ cun 2909  ⟨cop 3370   class class class wbr 3755 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-br 3756 This theorem is referenced by:  dmun  4485  qfto  4657  poleloe  4667  cnvun  4672  coundi  4765  coundir  4766  brdifun  6069  ltxrlt  6882  ltxr  8465
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