Theorem brun 4693

Description: The union of two binary relations. (Contributed by NM, 21-Dec-2008.)

Assertion

Ref	Expression
brun	⊢ (A(R ∪ S)B ↔ (ARB ∨ ASB))

Proof of Theorem brun

Step	Hyp	Ref	Expression
1		elun 3221	. 2 ⊢ (⟨A, B⟩ ∈ (R ∪ S) ↔ (⟨A, B⟩ ∈ R ∨ ⟨A, B⟩ ∈ S))
2		df-br 4641	. 2 ⊢ (A(R ∪ S)B ↔ ⟨A, B⟩ ∈ (R ∪ S))
3		df-br 4641	. . 3 ⊢ (ARB ↔ ⟨A, B⟩ ∈ R)
4		df-br 4641	. . 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))

Syntax hints:   ↔ wb 176   ∨ wo 357   ∈ wcel 1710   ∪ cun 3208  ⟨cop 4562   class class class wbr 4640

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-br 4641

This theorem is referenced by:  dmun  4913  cnvun  5034  coundi  5083  coundir  5084  nchoicelem16  6305