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Theorem brun 4692
 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 3220 . 2 (A, B (RS) ↔ (A, B R A, B S))
2 df-br 4640 . 2 (A(RS)BA, B (RS))
3 df-br 4640 . . 3 (ARBA, B R)
4 df-br 4640 . . 3 (ASBA, B S)
53, 4orbi12i 507 . 2 ((ARB ASB) ↔ (A, B R A, B S))
61, 2, 53bitr4i 268 1 (A(RS)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-1 5  ax-2 6  ax-3 7  ax-mp 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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