Theorem brun 4033

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 3263	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∪ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∨ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 3983	. 2 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∪ 𝑆))
3		df-br 3983	. . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4		df-br 3983	. . 3 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
5	3, 4	orbi12i 754	. 2 ⊢ ((𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∨ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
6	1, 2, 5	3bitr4i 211	1 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵))

Syntax hints:   ↔ wb 104   ∨ wo 698   ∈ wcel 2136   ∪ cun 3114  ⟨cop 3579   class class class wbr 3982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-br 3983

This theorem is referenced by:  dmun  4811  qfto  4993  poleloe  5003  cnvun  5009  coundi  5105  coundir  5106  brdifun  6528  ltxrlt  7964  ltxr  9711