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Mirrors > Home > MPE Home > Th. List > brdif | Structured version Visualization version GIF version |
Description: The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.) |
Ref | Expression |
---|---|
brdif | ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3839 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∖ 𝑆) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
2 | df-br 4930 | . 2 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ∖ 𝑆)) | |
3 | df-br 4930 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
4 | df-br 4930 | . . . 4 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
5 | 4 | notbii 312 | . . 3 ⊢ (¬ 𝐴𝑆𝐵 ↔ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆) |
6 | 3, 5 | anbi12i 617 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ ¬ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
7 | 1, 2, 6 | 3bitr4i 295 | 1 ⊢ (𝐴(𝑅 ∖ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ ¬ 𝐴𝑆𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ wa 387 ∈ wcel 2050 ∖ cdif 3826 〈cop 4447 class class class wbr 4929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-v 3417 df-dif 3832 df-br 4930 |
This theorem is referenced by: fundif 6236 fndmdif 6637 isocnv3 6908 brdifun 8118 dflt2 12358 pltval 17428 ltgov 26085 opeldifid 30115 qtophaus 30750 dftr6 32512 dffr5 32515 fundmpss 32535 slenlt 32758 brsset 32877 dfon3 32880 brtxpsd2 32883 dffun10 32902 elfuns 32903 dfrecs2 32938 dfrdg4 32939 dfint3 32940 brub 32942 broutsideof 33109 brvdif 34972 frege124d 39475 |
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