Theorem brun 3801

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 3078	. 2 ⊢ (⟨A, B⟩ ∈ (𝑅 ∪ 𝑆) ↔ (⟨A, B⟩ ∈ 𝑅 ∨ ⟨A, B⟩ ∈ 𝑆))
2		df-br 3756	. 2 ⊢ (A(𝑅 ∪ 𝑆)B ↔ ⟨A, B⟩ ∈ (𝑅 ∪ 𝑆))
3		df-br 3756	. . 3 ⊢ (A𝑅B ↔ ⟨A, B⟩ ∈ 𝑅)
4		df-br 3756	. . 3 ⊢ (A𝑆B ↔ ⟨A, B⟩ ∈ 𝑆)
5	3, 4	orbi12i 680	. 2 ⊢ ((A𝑅B ∨ A𝑆B) ↔ (⟨A, B⟩ ∈ 𝑅 ∨ ⟨A, B⟩ ∈ 𝑆))
6	1, 2, 5	3bitr4i 201	1 ⊢ (A(𝑅 ∪ 𝑆)B ↔ (A𝑅B ∨ A𝑆B))

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