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Mirrors > Home > MPE Home > Th. List > brun | Structured version Visualization version GIF version |
Description: The union of two binary relations. (Contributed by NM, 21-Dec-2008.) |
Ref | Expression |
---|---|
brun | ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑆𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 4113 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∪ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∨ ⟨𝐴, 𝐵⟩ ∈ 𝑆)) | |
2 | df-br 5111 | . 2 ⊢ (𝐴(𝑅 ∪ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∪ 𝑆)) | |
3 | df-br 5111 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅) | |
4 | df-br 5111 | . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 846 ∈ wcel 2107 ∪ cun 3913 ⟨cop 4597 class class class wbr 5110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-un 3920 df-br 5111 |
This theorem is referenced by: dmun 5871 qfto 6080 poleloe 6090 cnvun 6100 coundi 6204 coundir 6205 fununmo 6553 eqfunresadj 7310 brdifun 8684 fpwwe2lem12 10585 ltxrlt 11232 ltxr 13043 dfle2 13073 brprop 31653 satfbrsuc 34000 dfso2 34367 dfon3 34506 brcup 34553 dfrdg4 34565 dffrege99 42308 |
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