Theorem brun 5198

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 4162	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∪ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∨ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 5148	. 2 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∪ 𝑆))
3		df-br 5148	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 5148	. . 3 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	3, 4	orbi12i 914	. 2 ⊢ ((𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∨ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
6	1, 2, 5	3bitr4i 303	1 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵))

Syntax hints:   ↔ wb 206   ∨ wo 847   ∈ wcel 2105   ∪ cun 3960  ⟨cop 4636   class class class wbr 5147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-un 3967  df-br 5148

This theorem is referenced by:  dmun  5923  qfto  6143  poleloe  6153  cnvun  6164  coundi  6268  coundir  6269  fununmo  6614  eqfunresadj  7379  brdifun  8773  fpwwe2lem12  10679  ltxrlt  11328  ltxr  13154  dfle2  13185  brprop  32711  satfbrsuc  35350  dfso2  35734  dfon3  35873  brcup  35920  dfrdg4  35932  dffrege99  43951