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Mirrors > Home > MPE Home > Th. List > brsymdif | Structured version Visualization version GIF version |
Description: The binary relationship of a symmetric difference. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
brsymdif | ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4584 | . 2 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 △ 𝑆)) | |
2 | elsymdif 3811 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 △ 𝑆) ↔ ¬ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
3 | df-br 4584 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
4 | df-br 4584 | . . . 4 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
5 | 3, 4 | bibi12i 328 | . . 3 ⊢ ((𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
6 | 2, 5 | xchbinxr 324 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 △ 𝑆) ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
7 | 1, 6 | bitri 263 | 1 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∈ wcel 1977 △ csymdif 3805 〈cop 4131 class class class wbr 4583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-symdif 3806 df-br 4584 |
This theorem is referenced by: brtxpsd 31171 |
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