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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 5150 | . 2 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆)) | |
2 | elsymdif 4248 | . . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆) ↔ ¬ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)) | |
3 | df-br 5150 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅) | |
4 | df-br 5150 | . . . 4 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆) | |
5 | 3, 4 | bibi12i 340 | . . 3 ⊢ ((𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)) |
6 | 2, 5 | xchbinxr 335 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 △ 𝑆) ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
7 | 1, 6 | bitri 275 | 1 ⊢ (𝐴(𝑅 △ 𝑆)𝐵 ↔ ¬ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∈ wcel 2107 △ csymdif 4242 ⟨cop 4635 class class class wbr 5149 |
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 2704 |
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 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-un 3954 df-symdif 4243 df-br 5150 |
This theorem is referenced by: brtxpsd 34866 |
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