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Mirrors > Home > MPE Home > Th. List > brsymdif | Structured version Visualization version GIF version |
Description: Characterization of the symmetric difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
brsymdif | ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5031 | . 2 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 △ 𝑆)) | |
2 | elsymdif 4174 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 △ 𝑆) ↔ ¬ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
3 | df-br 5031 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
4 | df-br 5031 | . . . 4 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
5 | 3, 4 | bibi12i 343 | . . 3 ⊢ ((𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
6 | 2, 5 | xchbinxr 338 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 △ 𝑆) ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
7 | 1, 6 | bitri 278 | 1 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∈ wcel 2111 △ csymdif 4168 〈cop 4531 class class class wbr 5030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-dif 3884 df-un 3886 df-symdif 4169 df-br 5031 |
This theorem is referenced by: brtxpsd 33468 |
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