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Theorem brun 5113
 Description: The union of two binary relations. (Contributed by NM, 21-Dec-2008.)
Assertion
Ref Expression
brun (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))

Proof of Theorem brun
StepHypRef Expression
1 elun 4128 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∨ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2 df-br 5063 . 2 (𝐴(𝑅𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆))
3 df-br 5063 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4 df-br 5063 . . 3 (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
53, 4orbi12i 910 . 2 ((𝐴𝑅𝐵𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∨ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
61, 2, 53bitr4i 304 1 (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∨ wo 843   ∈ wcel 2107   ∪ cun 3937  ⟨cop 4569   class class class wbr 5062 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-v 3501  df-un 3944  df-br 5063 This theorem is referenced by:  dmun  5777  qfto  5978  poleloe  5988  cnvun  5998  coundi  6097  coundir  6098  fununmo  6397  brdifun  8311  fpwwe2lem13  10056  ltxrlt  10703  ltxr  12503  dfle2  12533  brprop  30349  satfbrsuc  32500  dfso2  32877  eqfunresadj  32891  dfon3  33240  brcup  33287  dfrdg4  33299  dffrege99  40176
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